Radii of γ-Spirallike of q-Special Functions

Abstract

The geometric properties of q-Bessel and q-Bessel-Struve functions are examined in this study. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For these normalized functions, the radii of $\gamma$-spirallike and convex $\gamma$-spirallike of order $\sigma$ are determined using their Hadamard factorization. These findings extend the known results for Bessel and Struve functions. The characterization of entire functions from the Laguerre-Pólya class plays an important role in our proofs. Additionally, the interlacing property of zeros of q-Bessel and q-Bessel-Struve functions and their derivatives is useful in the proof of our main theorems.

1. Introduction

Special functions have an indispensable role in many branches of mathematics and applied mathematics. Therefore, it is important to examine its properties from many aspects. In recent years, there has been intense interest in some special functions in terms of geometric function theory, especially q-special functions, which have an important place in the fields of mathematical physics and engineering.

The q-analogs are a technique employed in mathematical physics and mathematical analysis, crucial for accurately defining various physical and mathematical systems. Also, q-Bessel and q-Bessel-Struve functions hold significant importance within the geometric function theory. These functions exhibit similarities to classical Bessel and Struve functions but have been modified using q-analog techniques.

There is an extensive literature dealing with geometric properties of different types of special functions. For instance, in 2014, Baricz et al. [1], by considering a much simpler approach, succeeded in determining the radius of starlikeness of the normalized Bessel functions. In the same year, Baricz and Szász [2] obtained the radius of convexity of the normalized Bessel functions. Finding radii of starlikeness and convexity using the zeros of Bessel functions was later extended to other special functions. Their geometric properties such as starlikeness, convexity, close-to-convexity, uniformly convex, lemniscate starlikeness and convexity, Janowski starlikeness and convexity, $\gamma$-spirallike and convex $\gamma$-spirallike have been investigated by many mathematicians. For example, Bessel functions [1,2,3,4,5,6,7,8], Lommel and Struve functions [9,10,11,12], Wright functions [12,13,14], Mittag-Leffler functions [12,13,15,16,17,18], Ramanujan functions [12,17,19], Legendre polynomials [12,20], the function $N_{\nu }\left(z\right)=a{z}^2{J}_{\nu }^{″}\left(z\right)+bz{J}_{\nu }^{\prime }\left(z\right)+c{J}_{\nu }\left(z\right)$ [21,22,23,24,25] and q-special functions [6,7,26,27,28,29,30,31,32].

Building on the motivation from the literature discussing geometric properties of special functions, particularly in relation to q-Bessel and q-Bessel-Struve functions, I have explored the radii of $\gamma$-spirallike and convex $\gamma$-spirallike properties of order $\sigma .$ These properties are crucial in geometric function theory as they define the regions in which these functions map the unit disk.

In conclusion, the investigation of these geometric properties for q-Bessel and q-Bessel-Struve functions builds upon the rich theoretical foundation laid out in the literature, offering new perspectives and avenues for future research in mathematical analysis and its applications.

Under the normalization $\hslash \left(0\right)=0={\hslash }^{\prime }\left(0\right)-1$, let $\mathcal{A}$ show the analytic functions class in the unit disk $\mathsf{\Delta }:={\mathsf{\Delta }}_1$, where ${\mathsf{\Delta }}_r:=\{\omega \in \mathbb{C}:\left|\omega \right|<r\}$. We refer to that function $\hslash \in \mathcal{A}$ is $\gamma$-spirallike of order $\sigma$ if and only if

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega {\hslash }^{\prime }\left(\omega \right)}{\hslash \left(\omega \right)}}\right)>\sigma cos\gamma ,$$

where $\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ and $0\le \sigma <1$. We denote the class of functions like this ${\mathcal{S}}_p^{\gamma }\left(\sigma \right)$. We also denote its convex analog, that is the class ${\mathcal{CS}}_p^{\gamma }\left(\sigma \right)$ of convex $\gamma$-spirallike functions of order $\sigma$, which is defined below

$$\Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega {\hslash }^{″}\left(\omega \right)}{{\hslash }^{\prime }\left(\omega \right)}}\right)\right)>\sigma cos\gamma .$$

The class ${\mathcal{S}}_p^{\gamma }\left(0\right)$ was presented by Spacek [33]. Every function in ${\mathcal{CS}}_p^{\gamma }\left(\sigma \right)$ is univalent in $\mathsf{\Delta }$, although they need not be starlike. In addition, it is worth noting that for overall values of $\gamma \left(\right|\gamma |<\pi /2)$, a function in ${\mathcal{CS}}_p^{\gamma }\left(0\right)$ need not be univalent in $\mathsf{\Delta }$. For instance: $\hslash \left(\omega \right)=i{(1-\omega )}^i-i\in {\mathcal{CS}}_p^{\pi /4}\left(0\right)$, but not univalent. Actually, $\hslash \in {\mathcal{CS}}_p^{\gamma }\left(0\right)$ is univalent if $0<cos\gamma <1/2$, see Robertson [34] and Pfaltzgraff [35]. Note, that for $\gamma =0$, the classes ${\mathcal{S}}_p^{\gamma }\left(\sigma \right)$ and ${\mathcal{CS}}_p^{\gamma }\left(\sigma \right)$ reduce to the classes convex and starlike functions with order $\sigma$, given by

$$\Re \left({\displaystyle \frac{\omega {\hslash }^{\prime }\left(\omega \right)}{\hslash \left(\omega \right)}}\right)>\sigma \mathit{and}\Re \left(1+{\displaystyle \frac{\omega {\hslash }^{″}\left(\omega \right)}{{\hslash }^{\prime }\left(\omega \right)}}\right)>\sigma ,$$

which we denote this with ${\mathcal{S}}^{*}\left(\sigma \right)$ and $\mathcal{C}\left(\sigma \right)$, respectively.

Recently, linkages between special functions and their geometrical features have been demonstrated using radius problems [1,3,4,9,10,11,14,15,17,20,23,25]. In this manner, the behavior of the positive roots of a specific function, as well as the Laguerre-Pólya class, are clearly important. A self-mapping real entire function L of the real line is in the Laguerre-Pólya class $\mathcal{LP}$, if for $c,\beta ,{\omega }_k\in \mathbb{R}$, $a\ge 0$, $m\in \mathbb{N}\cup \left\{0\right\}$ we have

$$L\left(\omega \right)=c{\omega }^m{e}^{-a{\omega }^2+\beta \omega }\prod _{k\ge 1}\left(1+{\displaystyle \frac{\omega }{{\omega }_k}}\right)e^{-{\displaystyle \frac{\omega }{{\omega }_k}}}$$

with $\sum {{\omega }_k}^{-2}<\infty$, see [9], ([36] p. 703), [37] and the references therein. The class $\mathcal{LP}$ is of entire functions approximated uniformly on the complex plane’s compact sets by polynomials with only real zeros. This class is closed under differentiation.

We will also need to recall the following results for further development:

Lemma 1 

(see [38,39]). Take a look at the power series $\hslash \left(\omega \right)={\sum }_{n\ge 0}{a}_n{\omega }^n$ and $\ell \left(\omega \right)={\sum }_{n\ge 0}{b}_n{\omega }^n,$ where $a_n\in \mathbb{R}$ and $b_n>0$ for all $n\ge 0.$ Assume that both series converge on $(-r,r),$ for some $r>0.$ If the sequence ${\left\{a_n/b_n\right\}}_{n\ge 0}$ is decreasing (increasing), then the function $\omega \mapsto \hslash \left(\omega \right)/\ell \left(\omega \right)$ is decreasing (increasing) too on $\left(0,r\right).$ This is true for the power series

$$\hslash \left(\omega \right)=\sum _{n\ge 0}{a}_n{\omega }^{2n}\mathit{and}\ell \left(\omega \right)=\sum _{n\ge 0}{b}_n{\omega }^{2n}.$$

The ${\mathcal{S}}^{*}\left(\sigma \right)$-radius, which is given below

$$sup\{r\in {\mathbb{R}}^{+}:\Re \left({\displaystyle \frac{\omega {\hslash }^{\prime }\left(\omega \right)}{\hslash \left(\omega \right)}}\right)>\sigma ,\omega \in {\mathsf{\Delta }}_r\}$$

similarly, the $\mathcal{C}\left(\sigma \right)$-radius has recently been obtained for some normalized forms of Bessel functions in [1,3,4] (Watson’s treatise [40] is an excellent resource on Bessel function), Struve functions studied in [9,11], Wright functions in [14], Lommel functions radii properties studied in [9,11], Legendre polynomials of odd degree radii results deal in [20] and recently, Ramanujan type entire functions were studied in [19]. These function’s radii problems collectively for unified subclasses of starlike and convex functions have been studied in [17].

According to the literature review, ${\mathcal{S}}_p^{\gamma }\left(\sigma \right)$-radius and ${\mathcal{CS}}_p^{\gamma }\left(\sigma \right)$-radius for special functions have not been discussed so far. So, in this study, we now aim to derive the radius of $\gamma$-spirallike of order $\sigma$, which is provided below

$$r_{sp}^{*}(\sigma ,\gamma ;\hslash )=sup\left\{r\in {\mathbb{R}}^{+}:\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega {\hslash }^{\prime }\left(\omega \right)}{\hslash \left(\omega \right)}}\right)>\sigma cos\gamma ,\left|\gamma \right|<{\displaystyle \frac{\pi }{2}},\omega \in {\mathsf{\Delta }}_r\right\}$$

and also the radius of convex $\gamma$-spirallike of order $\sigma$, which is

$$r_{sp}^c(\sigma ,\gamma ;\hslash )=sup\left\{r\in {\mathbb{R}}^{+}:\Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega {\hslash }^{″}\left(\omega \right)}{{\hslash }^{\prime }\left(\omega \right)}}\right)\right)>\sigma cos\gamma ,\left|\gamma \right|<{\displaystyle \frac{\pi }{2}},\omega \in {\mathsf{\Delta }}_r\right\}$$

for the function ℏ in $\mathcal{A}$ to be a special function.

2. $\mathit{q}$-Bessel Functions

The infinite series representation of the first kind of Bessel function [40] is defined as follows:

$$J_{\tau }\left(\omega \right)=\sum _{k\ge 0}{\displaystyle \frac{{\left(-1\right)}^k}{k!\mathsf{\Gamma }(k+\tau +1)}}{\left({\displaystyle \frac{\omega }{2}}\right)}^{2k+\tau },$$

where $\omega \in \mathbb{C}$ and $\tau \in \mathbb{C}$ such that $\tau \ne -1,-2,\dots .$

Lommel’s well-known conclusion represents that if $\tau >-1,$ then the zeros of the Bessel function $J_{\tau }$ are all real. Thus, if $j_{\tau ,k}$ and $j_{\tau ,k}^{\prime }$ denote the k-th positive zeros of $J_{\tau }$ and $J_{\tau }^{\prime },$ respectively, then the Bessel function and its derivative admit the Weierstrassian decomposition of the forms [40]

$$J_{\tau }\left(\omega \right)={\displaystyle \frac{{\omega }^{\tau }}{2^{\tau }\mathsf{\Gamma }\left(\tau +1\right)}}\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^2}}\right)$$

(1)

and

$$J_{\tau }^{\prime }\left(\omega \right)={\displaystyle \frac{{\omega }^{\tau -1}}{2^{\tau }\mathsf{\Gamma }\left(\tau +1\right)}}\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^{\prime 2}}}\right).$$

(2)

The convergence is uniform on each compact subset of $\mathbb{C}$ for the above infinite products. The Jackson and Hahn-Exton q-Bessel functions are clearly stated by

$$J^{\left(2\right)}(\omega ;q)={\displaystyle \frac{{\left(q^{\tau +1};q\right)}_{\infty }}{{\left(q;q\right)}_{\infty }}}\sum _{k\ge 0}{\displaystyle \frac{{\left(-1\right)}^k{q}^{k(k+\tau )}}{{\left(q;q\right)}_k{\left(q^{\tau +1};q\right)}_k}}{\left({\displaystyle \frac{\omega }{2}}\right)}^{2k+\tau }$$

and

$$J^{\left(3\right)}(\omega ;q)={\displaystyle \frac{{\left(q^{\tau +1};q\right)}_{\infty }}{{\left(q;q\right)}_{\infty }}}\sum _{k\ge 0}{\displaystyle \frac{{\left(-1\right)}^k{q}^{\frac{1}{2}k(k+1)}}{{\left(q;q\right)}_k{\left(q^{\tau +1};q\right)}_k}}{\omega }^{2k+\tau },$$

where $\omega \in \mathbb{C},\tau >-1,q\in \left(0,1\right)$ and

$${\left(a;q\right)}_0=1,{\left(a;q\right)}_k=\prod _{n=1}^k\left(1-a{q}^{n-1}\right),{\left(a;q\right)}_{\infty }=\prod _{k\ge 1}\left(1-a{q}^{k-1}\right).$$

Particularly, these analytic functions are q-extensions of $J_{\tau }.$ Namely, for fixed $\omega$ we have $J^{\left(2\right)}\left((1-\omega )q;q\right)⟶J_{\tau }\left(\omega \right)$ and $J^{\left(3\right)}\left((1-\omega )q;q\right)⟶J_{\tau }\left(2\omega \right)$ as $q\nearrow 1.$ Watson’s treatise [40] is an excellent resource for Bessel functions. Recent developments on Bessel and its extension can be found in [41,42,43,44] and the references therein.

In this section, we investigate the geometric properties of the normalized Jackson and Hahn-Exton q-Bessel functions. Combining the methods deployed in [1,2,9] we determine precisely the radii of $\gamma$-spirallike of order $\sigma$ and convex $\gamma$-spirallike of order $\sigma$ for each of the six functions. In between the proofs, we exercise the simple properties of the Laguerre-Pólya class, as well as the interlacing property of the zeros of the Jackson and Hahn-Exton q-Bessel functions.

The current study shows that there is no important difference between Jackson and Hahn-Exton q-Bessel functions when treating the problem about the radii of $\gamma$-spirallike of order $\sigma$ and convex $\gamma$-spirallike of order $\sigma$. As a result, one may predict that the other geometric features of these two q-extensions of Bessel’s function are comparable.

Observe that neither $J_{\tau }^{\left(2\right)}(\omega ;q)$, nor $J_{\tau }^{\left(3\right)}(\omega ;q)$ belongs to $\mathcal{A}$, and therefore, $\tau >-1$ we define three normalized functions that stem from $J_{\tau }^{\left(2\right)}(\omega ;q)$:

$$\begin{array}{ccc}\hfill f_{\tau }^{\left(2\right)}(\omega ;q)& =& {\left(2^{\tau }{c}_{\tau }\left(q\right)J_{\tau }^{\left(2\right)}(\omega ;q)\right)}^{{\displaystyle \frac{1}{\tau }}},\tau \ne 0,\hfill \\ \hfill g_{\tau }^{\left(2\right)}(\omega ;q)& =& 2^{\tau }{c}_{\tau }\left(q\right){\omega }^{1-\tau }{J}_{\tau }^{\left(2\right)}(\omega ;q),\hfill \\ \hfill h_{\tau }^{\left(2\right)}(\omega ;q)& =& 2^{\tau }{c}_{\tau }\left(\omega \right){\omega }^{1-{\displaystyle \frac{\tau }{2}}}{J}_{\tau }^{\left(2\right)}(\sqrt{\omega };q)\hfill \end{array}$$

(3)

where $c_{\tau }\left(q\right)={\left(q;q\right)}_{\infty }/{\left(q^{\tau +1};q\right)}_{\infty }.$ Similarly,

$$\begin{array}{ccc}\hfill f_{\tau }^{\left(3\right)}(\omega ;q)& =& {\left(c_{\tau }\left(q\right)J_{\tau }^{\left(3\right)}(\omega ;q)\right)}^{{\displaystyle \frac{1}{\tau }}},\tau \ne 0,\hfill \\ \hfill g_{\tau }^{\left(3\right)}(\omega ;q)& =& c_{\tau }\left(q\right){\omega }^{1-\tau }{J}_{\tau }^{\left(3\right)}(\omega ;q),\hfill \\ \hfill h_{\tau }^{\left(3\right)}(\omega ;q)& =& c_{\tau }\left(\omega \right){\omega }^{1-{\displaystyle \frac{\tau }{2}}}{J}_{\tau }^{\left(3\right)}(\sqrt{\omega };q).\hfill \end{array}$$

(4)

Clearly, the functions $f_{\tau }^{\left(s\right)}(\omega ;q),g_{\tau }^{\left(s\right)}(\omega ;q),h_{\tau }^{\left(s\right)}(\omega ;q),s\in \left\{2,3\right\},$ belong to the class $\mathcal{A}.$ The primary rationale for considering these six functions stems from their connections to the limiting cases found in the literature concerning Bessel functions, as discussed in [16] and the associated references.

Lemma 2 

([28]). If $\tau >-1,$ then $\omega \mapsto {\mathcal{J}}_{\tau }^{\left(2\right)}(\omega ;q)=2^{\tau }{c}_{\tau }\left(q\right){\omega }^{-\tau }{J}_{\tau }^{\left(2\right)}(\omega ;q)$ and $\omega \mapsto {\mathcal{J}}_{\tau }^{\left(3\right)}(\omega ;q)=c_{\tau }\left(q\right){\omega }^{-\tau }{J}_{\tau }^{\left(3\right)}(\omega ;q)$ are entire functions of order $\rho =0$. Consequently, their Hadamard factorization for $\omega \in \mathbb{C}$ are of the form

$${\mathcal{J}}_{\tau }^{\left(2\right)}(\omega ;q)=\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^2\left(q\right)}}\right)\mathit{and}{\mathcal{J}}_{\tau }^{\left(3\right)}(\omega ;q)=\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\eta }_{\tau ,k}^2\left(q\right)}}\right),$$

(5)

where $j_{\tau ,k}\left(q\right)$ and ${\eta }_{\tau ,k}\left(q\right)$ are the k-th positive zeros of the functions $J_{\tau }^{\left(2\right)}(\omega ;q)$ and $J_{\tau }^{\left(3\right)}(\omega ;q).$

Lemma 3 

([28]). If $\tau >0,$ then $\omega \mapsto (2^{\tau }/\tau )c_{\tau }\left(q\right){\omega }^{1-\tau }d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega$ and $\omega \mapsto (1/\tau )c_{\tau }\left(q\right){\omega }^{1-\tau }d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega$ are entire functions of order $\rho =0$. Consequently, their Hadamard factorization for $\omega \in \mathbb{C}$ are of the form

$$d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega ={\displaystyle \frac{\tau {\omega }^{\tau -1}}{2^{\tau }{c}_{\tau }\left(q\right)}}\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^{\prime 2}\left(q\right)}}\right)$$

(6)

and

$$d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega ={\displaystyle \frac{\tau {\omega }^{\tau -1}}{c_{\tau }\left(q\right)}}\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\eta }_{\tau ,k}^{\prime 2}\left(q\right)}}\right),$$

(7)

where $j_{\tau ,k}^{\prime }\left(q\right)$ and ${\eta }_{\tau ,k}^{\prime }\left(q\right)$ are the k-th positive zeros of $\omega \mapsto d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega$ and $\omega \mapsto d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega .$

Lemma 4 

([28]). Let $\tau >-1$ and $a<0.$ Then the functions $\omega \mapsto \left(2a+\tau \right)$$J_{\tau }^{\left(2\right)}(\omega ;q)-\omega ·d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega$ and $\omega \mapsto \left(2a+\tau \right)J_{\tau }^{\left(3\right)}(\omega ;q)-\omega ·d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega$ can be expressed in the following way

$$c_{\tau }\left(q\right)\left(\omega \mapsto \left(2a+\tau \right)J_{\tau }^{\left(2\right)}(\omega ;q)-\omega ·d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega \right)=2{\left({\displaystyle \frac{\omega }{2}}\right)}^{\tau }{\varphi }_{\tau }(\omega ;q),$$

$$c_{\tau }\left(q\right)\left(\omega \mapsto \left(2a+\tau \right)J_{\tau }^{\left(3\right)}(\omega ;q)-\omega ·d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega \right)=2{\left(\omega \right)}^{\tau }{\psi }_{\tau }(\omega ;q),$$

where ${\varphi }_{\tau }(\omega ;q)$ and ${\psi }_{\tau }(\omega ;q)$ are entire functions from the Laguerre-Pólya class $\mathcal{LP}.$ Furthermore, the lowest positive zero of ${\varphi }_{\tau }(\omega ;q)$ does not exceed the first positive zero $j_{\tau ,1}\left(q\right),$ and the lowest positive zero of ${\psi }_{\tau }(\omega ;q)$ is smaller than ${\eta }_{\tau ,1}\left(q\right).$

Lemma 5 

([28]). Between any two consecutive roots of $\omega \mapsto J_{\tau }^{\left(s\right)}(\omega ;q)$ the function $\omega \mapsto d{J}_{\tau }^{\left(s\right)}(\omega ;q)/d\omega$ has precisely one zero when $\tau \ge 0$ and $s\in \left\{2,3\right\}.$

The first principal result gives radii of $\gamma$-spirallike of order $\sigma$.

Theorem 1. 

Let $\sigma \in [0,1),$$\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ and $s\in \left\{2,3\right\}$. The following statements are true:

(i)

If $\tau >0,$ then the radius $r_{sp}^{*}(\sigma ,\gamma ;f_{\tau })$ is the lowest positive roots of the equation

$$r·d{J}_{\tau }^{\left(s\right)}(r;q)/dr-\left[1-\left(1-\sigma \right)cos\gamma \right]\tau J_{\tau }^{\left(s\right)}(r;q)=0.$$

Moreover, if $\tau \in \left(-1,0\right),$ then the radius $r_{sp}^{*}(\sigma ,\gamma ;f_{\tau })$ is the lowest positive roots of the equation

$$ir·d{J}_{\tau }^{\left(s\right)}(ir;q)/dr-\left[1-\left(1-\sigma \right)cos\gamma \right]\tau J_{\tau }^{\left(s\right)}(ir;q)=0.$$

(ii)

If $\tau >-1,$ then the radius $r_{sp}^{*}(\sigma ,\gamma ;g_{\tau })$ is the lowest positive roots of the equation

$$r·d{J}_{\tau }^{\left(s\right)}(r;q)/dr-\left[\tau -\left(1-\sigma \right)cos\gamma \right]J_{\tau }^{\left(s\right)}(r;q)=0.$$

(iii)

If $\tau >-1,$ then the radius $r_{sp}^{*}(\sigma ,\gamma ;h_{\tau })$ is the lowest positive roots of the equation

$$\sqrt{r}·d{J}_{\tau }^{\left(s\right)}(\sqrt{r};q)/dr-\left[\tau -2\left(1-\sigma \right)cos\gamma \right]J_{\tau }^{\left(s\right)}(\sqrt{r};q)=0.$$

Proof. 

The proofs for the cases $s=2$ and $s=3$ are nearly identical; the only difference is that the zeros $j_{\tau ,k}\left(q\right)$ and ${\eta }_{\tau ,k}\left(q\right)$ in the proofs are different. As a result, we will only present the proof for the case $s=2$, and for clarity, we will use the notations: $J_{\tau }(\omega ;q)=J_{\tau }^{\left(2\right)}(\omega ;q),$$f_{\tau }(\omega ;q)=f_{\tau }^{\left(2\right)}(\omega ;q),$$g_{\tau }(\omega ;q)=g_{\tau }^{\left(2\right)}(\omega ;q),$$h_{\tau }(\omega ;q)=h_{\tau }^{\left(2\right)}(\omega ;q)$ and $J_{\tau }^{\prime }(\omega ;q)=d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega .$

First we prove part $\left(i\right)$ for $\tau >0$ and parts $\left(ii\right)$ and $\left(iii\right)$ for $\tau >-1.$ We need to show that for $\sigma \in [0,1)$ and $\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ the inequalities

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega f_{\tau }^{\prime }(\omega ;q)}{f_{\tau }(\omega ;q)}}\right)>\sigma cos\gamma ,\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega g_{\tau }^{\prime }(\omega ;q)}{g_{\tau }(\omega ;q)}}\right)>\sigma cos\gamma$$

(8)

and

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega h_{\tau }^{\prime }(\omega ;q)}{h_{\tau }(\omega ;q)}}\right)>\sigma cos\gamma$$

(9)

are valid for $\omega \in {\mathsf{\Delta }}_{r_{sp}^{*}(\sigma ,\gamma ;f_{\tau })},\omega \in {\mathsf{\Delta }}_{r_{sp}^{*}(\sigma ,\gamma ;g_{\tau })}$ and $\omega \in {\mathsf{\Delta }}_{r_{sp}^{*}(\sigma ,\gamma ;h_{\tau })}$ accordingly, and none of the disparities listed above apply in larger disks.

Using (5), we obtain after the logarithmic differentiation:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\omega f_{\tau }^{\prime }(\omega ;q)}{f_{\tau }(\omega ;q)}}& =& {\displaystyle \frac{1}{\tau }}{\displaystyle \frac{\omega J_{\tau }^{\prime }(\omega ;q)}{J_{\tau }(\omega ;q)}}=1-{\displaystyle \frac{1}{\tau }}\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}},\hfill \\ \hfill {\displaystyle \frac{\omega g_{\tau }^{\prime }(\omega ;q)}{g_{\tau }(\omega ;q)}}& =& 1-\tau +{\displaystyle \frac{\omega J_{\tau }^{\prime }(\omega ;q)}{J_{\tau }(\omega ;q)}}=1-\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}},\hfill \\ \hfill {\displaystyle \frac{\omega h_{\tau }^{\prime }(\omega ;q)}{h_{\tau }(\omega ;q)}}& =& 1-{\displaystyle \frac{\tau }{2}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\omega }{J}_{\tau }^{\prime }(\sqrt{\omega };q)}{J_{\tau }(\sqrt{\omega };q)}}=1-\sum _{k\ge 1}{\displaystyle \frac{\omega }{j_{\tau ,k}^2\left(q\right)-\omega }}.\hfill \end{array}$$

(10)

It is known [1] that if $\omega \in \mathbb{C}$ and $\lambda \in \mathbb{R}$ are such that $\left|\omega \right|\le r<\lambda ,$ then

$$\Re \left({\displaystyle \frac{\omega }{\lambda -\omega }}\right)\le \left|{\displaystyle \frac{\omega }{\lambda -\omega }}\right|\le {\displaystyle \frac{\left|\omega \right|}{\lambda -\left|\omega \right|}}.$$

(11)

Then, the inequality

$$\Re \left({\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}}\right)\le \left|{\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}}\right|\le {\displaystyle \frac{{\left|\omega \right|}^2}{j_{\tau ,k}^2\left(q\right)-{\left|\omega \right|}^2}}$$

holds for every $\left|\omega \right|<j_{\tau ,1}\left(q\right),$ which in turn implies that

$$\begin{array}{ccc}\hfill \Re \left(e^{-i\gamma }{\displaystyle \frac{\omega f_{\tau }^{\prime }(\omega ;q)}{f_{\tau }(\omega ;q)}}\right)& =& \Re \left(e^{-i\gamma }\right)-{\displaystyle \frac{1}{\tau }}\Re \left(e^{-i\gamma }\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}}\right)\hfill \\ & \ge & cos\gamma -{\displaystyle \frac{1}{\tau }}\sum _{k\ge 1}{\displaystyle \frac{2{\left|\omega \right|}^2}{j_{\tau ,k}^2\left(q\right)-{\left|\omega \right|}^2}}={\displaystyle \frac{\left|\omega \right|f_{\tau }^{\prime }(\left|\omega \right|;q)}{f_{\tau }(\left|\omega \right|;q)}}+cos\gamma -1,\hfill \end{array}$$

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega g_{\tau }^{\prime }(\omega ;q)}{g_{\tau }(\omega ;q)}}\right)\ge {\displaystyle \frac{\left|\omega \right|g_{\tau }^{\prime }(\left|\omega \right|;q)}{g_{\tau }(\left|\omega \right|;q)}}+cos\gamma -1$$

and

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega h_{\tau }^{\prime }(\omega ;q)}{h_{\tau }(\omega ;q)}}\right)\ge {\displaystyle \frac{\left|\omega \right|h_{\tau }^{\prime }(\left|\omega \right|;q)}{h_{\tau }(\left|\omega \right|;q)}}+cos\gamma -1$$

with equality when $\omega =\left|\omega \right|=r.$ Because of the latter inequalities and the minimum principle for harmonic functions, the corresponding inequalities in (8) and (9) hold if and only if $\left|\omega \right|<x_{\tau ,\sigma ,\gamma ,1},$$\left|\omega \right|<y_{\tau ,\sigma ,\gamma ,1}$ and $\left|\omega \right|<z_{\tau ,\sigma ,\gamma ,1},$ respectively, where $x_{\tau ,\sigma ,\gamma ,1},$$y_{\tau ,\sigma ,\gamma ,1}$ and $z_{\tau ,\sigma ,\gamma ,1}$ are the lowest positive roots of the equations

$${\displaystyle \frac{r{f}_{\tau }^{\prime }(r;q)}{f_{\tau }(r;q)}}=1-\left(1-\sigma \right)cos\gamma ,{\displaystyle \frac{r{g}_{\tau }^{\prime }(r;q)}{g_{\tau }(r;q)}}=1-\left(1-\sigma \right)cos\gamma$$

and

$${\displaystyle \frac{r{h}_{\tau }^{\prime }(r;q)}{h_{\tau }(r;q)}}=1-\left(1-\sigma \right)cos\gamma .$$

Because their solutions correspond to the zeros of the functions

$$r\mapsto r{J}_{\tau }^{\prime }(r;q)-\left[1-\left(1-\sigma \right)cos\gamma \right]\tau J_{\tau }(r;q),$$

$$r\mapsto r{J}_{\tau }^{\prime }(r;q)-\left[\tau -\left(1-\sigma \right)cos\gamma \right]J_{\tau }(r;q)$$

and

$$r\mapsto \sqrt{r}{J}_{\tau }^{\prime }(\sqrt{r};q)-\left[\tau -2\left(1-\sigma \right)cos\gamma \right]J_{\tau }(\sqrt{r};q),$$

the required result is obtained by substituting ${\displaystyle \frac{\tau \left(\sigma -1\right)cos\gamma }{2}},$${\displaystyle \frac{\left(\sigma -1\right)cos\gamma }{2}}$ and $\tau \left(\sigma -1\right)cos\gamma ,$ for a in Lemma 4, respectively. In another saying, Lemma 4 demonstrates that all of the above three functions have real zeros and their first positive zeros do not exceed the first positive zeros $j_{\tau ,1}\left(q\right),\sqrt{j_{\tau ,1}\left(q\right)}$, respectively. This assures that the aforementioned inequalities hold, so completing the proof of part $\left(i\right)$ when $\tau >0$, and parts $\left(ii\right)$ and $\left(iii\right)$ when $\tau >-1.$

To demonstrate the statement for component $\left(i\right)$ when $\tau \in (-1,0),$ first observe that the counterpart of (11), that is,

$$\Re \left({\displaystyle \frac{\omega }{\lambda -\omega }}\right)\ge {\displaystyle \frac{-\left|\omega \right|}{\lambda +\left|\omega \right|}},$$

(12)

is valid for all $\lambda \in \mathbb{R}$ and $\omega \in \mathbb{C}$ such that $\left|\omega \right|<\lambda .$ By using (12), for all $\tau >-1,k\in \left\{1,2,\dots \right\}$ and $\omega \in {\mathsf{\Delta }}_{j_{\tau ,1}}$ the inequality

$$\Re \left({\displaystyle \frac{{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}}\right)\ge {\displaystyle \frac{-{\left|\omega \right|}^2}{j_{\tau ,k}^2\left(q\right)+{\left|\omega \right|}^2}}$$

which holds for every $\left|\omega \right|<j_{\tau ,1}\left(q\right),$ and it implies that

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega f_{\tau }^{\prime }(\omega ;q)}{f_{\tau }(\omega ;q)}}\right)\ge cos\gamma +{\displaystyle \frac{1}{\tau }}\sum _{k\ge 1}{\displaystyle \frac{2{\left|\omega \right|}^2}{j_{\tau ,k}^2\left(q\right)+{\left|\omega \right|}^2}}={\displaystyle \frac{i\left|\omega \right|f_{\tau }^{\prime }(i\left|\omega \right|;q)}{f_{\tau }(i\left|\omega \right|;q)}}+cos\gamma -1.$$

In this situation, equality occurs if $\omega =i\left|\omega \right|=ir.$ Furthermore, the last inequality explains that

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega f_{\tau }^{\prime }(\omega ;q)}{f_{\tau }(\omega ;q)}}\right)\ge \sigma cos\gamma$$

if and only if $\left|\omega \right|<x_{\tau ,\sigma ,\gamma },$ where $x_{\tau ,\sigma ,\gamma }$ denotes the equation’s lowest positive root

$${\displaystyle \frac{ir{f}_{\tau }^{\prime }(ir;q)}{f_{\tau }(ir;q)}}=1-\left(1-\sigma \right)cos\gamma .$$

As a result of Lemma 4 the first positive zero of $\omega \mapsto i\omega J_{\tau }^{\prime }(i\omega ;q)-\left[1-\left(1-\sigma \right)cos\gamma \right]\tau J_{\tau }(i\omega ;q)$ does not exceed $j_{\tau ,1}\left(q\right)$ ensuring that the preceding inequalities are satisfied. We just need to demonstrate that the function mentioned above only has one zero in $(0,\infty ).$ Take note that, according to Lemma 1, the function

$$r\mapsto {\displaystyle \frac{ir{J}_{\tau }^{\prime }(ir;q)}{J_{\tau }(ir;q)}}={\displaystyle \frac{{\sum }_{k\ge 0}\frac{\left(2k+\tau \right)q^{k(k+\tau )}}{{\left(q;q\right)}_k{\left(q^{\tau +1};q\right)}_k}{\left(\frac{r}{2}\right)}^{2k+\tau }}{{\sum }_{k\ge 0}\frac{q^{k(k+\tau )}}{{\left(q;q\right)}_k{\left(q^{\tau +1};q\right)}_k}{\left(\frac{r}{2}\right)}^{2k+\tau }}}$$

is increasing on $(0,\infty )$ as a quotient of two power series whose positive coefficients form the increasing “quotient sequence” ${\left\{2k+\tau \right\}}_{n\ge 0}.$ When $r\mapsto 0,$ the above function tends to $\tau$, so its graph can only intersect the horizontal line $y=\left(1-\left(1-\sigma \right)cos\gamma \right)\tau >\tau$ once. This completes the proof of part $\left(i\right)$ of the theorem for $\tau \in (-1,0).$ □

Remark 1. 

Taking $\gamma =0$ in Theorem 1 yields ([28] Theorem 1).

Our second result in this section concerns the radii of convex $\gamma$-spirallike of order $\sigma .$

Theorem 2. 

Let $\sigma \in [0,1),$$\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ and $s\in \left\{2,3\right\}$.

(i)

If $\tau >0,$ then the radius $r_{sp}^c(\sigma ,\gamma ;f_{\tau })$ is the lowest positive roots of the equation

$${\displaystyle \frac{r·d^2{J}_{\tau }^{\left(s\right)}(r;q)/d{r}^2}{d{J}_{\tau }^{\left(s\right)}(r;q)/dr}}+\left({\displaystyle \frac{1}{\tau }}-1\right){\displaystyle \frac{r·d{J}_{\tau }^{\left(s\right)}(r;q)/dr}{J_{\tau }(r;q)}}=\left(\sigma -1\right)cos\gamma .$$

(ii)

If $\tau >-1,$ then the radius $r_{sp}^c(\sigma ,\gamma ;g_{\tau })$ is the lowest positive roots of the equation

$${\displaystyle \frac{r^2·d^2{J}_{\tau }^{\left(s\right)}(r;q)/d{r}^2+\left(2-\tau \right)r·d{J}_{\tau }^{\left(s\right)}(r;q)/dr}{r·d{J}_{\tau }^{\left(s\right)}(r;q)/dr+\left(1-\tau \right)J_{\tau }^{\left(s\right)}(r;q)}}=\tau +\left(\sigma -1\right)cos\gamma .$$

(iii)

If $\tau >-1,$ then the radius $r_{sp}^c(\sigma ,\gamma ;h_{\tau })$ is the lowest positive roots of the equation

$${\displaystyle \frac{r·d^2{J}_{\tau }^{\left(s\right)}(\sqrt{r};q)/d{r}^2+\left(3-\tau \right)\sqrt{r}·d{J}_{\tau }^{\left(s\right)}(\sqrt{r};q)/dr}{\sqrt{r}·d{J}_{\tau }^{\left(s\right)}(\sqrt{r};q)/dr+\left(2-\tau \right)J_{\tau }^{\left(s\right)}(\sqrt{r};q)}}=\tau +2\left(\sigma -1\right)cos\gamma .$$

Proof. 

The proofs for the conditions $s=2$ and $s=3$ are nearly identical; the only variation is that the zeros in the proofs are different. As in the last argument, we will explain the proof just for the case $s=2$, and for clarity, we will employ the following notations: $J_{\tau }(\omega ;q)=J_{\tau }^{\left(2\right)}(\omega ;q),$$f_{\tau }(\omega ;q)=f_{\tau }^{\left(2\right)}(\omega ;q),$$g_{\tau }(\omega ;q)=g_{\tau }^{\left(2\right)}(\omega ;q),$$h_{\tau }(\omega ;q)=h_{\tau }^{\left(2\right)}(\omega ;q),$$J_{\tau }^{\prime }(\omega ;q)=d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega .$ and $J_{\tau }^{\prime \prime }(\omega ;q)=d^2{J}_{\tau }^{\left(2\right)}(\omega ;q)/d{\omega }^2.$

(i)

Since

$$1+{\displaystyle \frac{\omega f_{\tau }^{″}(\omega ;q)}{f_{\tau }^{\prime }(\omega ;q)}}=1+{\displaystyle \frac{\omega J_{\tau }^{″}(\omega ;q)}{J_{\tau }^{\prime }(\omega ;q)}}+\left({\displaystyle \frac{1}{\tau }}-1\right){\displaystyle \frac{\omega J_{\tau }^{\prime }(\omega ;q)}{J_{\tau }(\omega ;q)}}$$

and by means of (6) and (7) we have

$$1+{\displaystyle \frac{\omega f_{\tau }^{″}(\omega ;q)}{f_{\tau }^{\prime }(\omega ;q)}}=1-\left({\displaystyle \frac{1}{\tau }}-1\right)\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}}-\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^{\prime 2}\left(q\right)-{\omega }^2}}.$$

For $0<\tau \le 1,$ by using the inequality (11), for all $\omega \in {\mathsf{\Delta }}_{j_{\tau ,1}^{\prime }\left(q\right)}$ we obtain the inequality

$$\begin{array}{cc}\hfill & \Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega f_{\tau }^{\prime \prime }(\omega ;q)}{f_{\tau }^{\prime }(\omega ;q)}}\right)\right)\hfill \\ \hfill & =\Re \left(e^{-i\gamma }\right)-\Re \left(e^{-i\gamma }\left(\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^{\prime 2}\left(q\right)-{\omega }^2}}+\left({\displaystyle \frac{1}{\tau }}-1\right)\sum _{k\ge 1}{\displaystyle \frac{2{\omega }^2}{j_{\tau ,k}^2\left(q\right)-{\omega }^2}}\right)\right)\hfill \\ \hfill & \ge cos\gamma -\left({\displaystyle \frac{1}{\tau }}-1\right)\sum _{k\ge 1}{\displaystyle \frac{2r^2}{j_{\tau ,k}^2\left(q\right)-r^2}}-\sum _{k\ge 1}{\displaystyle \frac{2r^2}{j_{\tau ,k}^{\prime 2}\left(q\right)-r^2}}\hfill \\ \hfill & \ge cos\gamma +{\displaystyle \frac{r{f}_{\tau }^{\prime \prime }(r;q)}{f_{\tau }^{\prime }(r;q)}}\hfill \end{array}$$

(13)

where $\left|\omega \right|=r$. Furthermore, notice that if we use the inequality ([2] Lemma 2.1)

$$\mu \Re \left({\displaystyle \frac{\omega }{a-\omega }}\right)-\Re \left({\displaystyle \frac{\omega }{b-\omega }}\right)\ge \mu {\displaystyle \frac{\left|\omega \right|}{a-\left|\omega \right|}}-{\displaystyle \frac{\left|\omega \right|}{b-\left|\omega \right|}}$$

(14)

where $a>b>0,$$\mu \in [0,1]$ and $\omega \in \mathbb{C}$ such that $\left|\omega \right|<b$, then we obtain that the inequality (13) is also correct for $\tau >1$. Here, we used the zeros $j_{\tau ,k}\left(q\right)$ and $j_{\tau ,k}^{\prime }\left(q\right)$ interlace according to Lemma 5. The inequality (13) implies for $r\in (0,j_{\tau ,1}^{\prime }\left(q\right))$

$$\underset{\omega \in {\mathsf{\Delta }}_r}{inf}\left\{\Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega f_{\tau }^{\prime \prime }(\omega ;q)}{f_{\tau }^{\prime }(\omega ;q)}}\right)-\sigma cos\gamma \right)\right\}=\left(1-\sigma \right)cos\gamma +{\displaystyle \frac{r{f}_{\tau }^{″}(r;q)}{f_{\tau }^{\prime }(r;q)}}.$$

Now, we define the function ${\mathsf{\Delta }}_{\tau }(r;q):\left(0,j_{\tau ,1}^{\prime }\left(q\right)\right)\to \mathbb{R},$

$${\mathsf{\Delta }}_{\tau }(r;q)=\left(1-\sigma \right)cos\gamma +{\displaystyle \frac{r{f}_{\tau }^{″}(r;q)}{f_{\tau }^{\prime }(r;q)}}$$

which is strictly decreasing since ${\mathsf{\Delta }}_{\tau }^{\prime }(r;q)<0$ for $\tau >0$ and $r\in \left(0,j_{\tau ,1}^{\prime }\left(q\right)\right).$ Observe also that ${lim}_{r\searrow 0}{\mathsf{\Delta }}_{\tau }(r;q)=\left(1-\sigma \right)cos\gamma >0$ and ${lim}_{r\nearrow j_{\tau ,1}^{\prime }\left(q\right)}{\mathsf{\Delta }}_{\tau }(r;q)=-\infty ,$ which means that for $\omega \in {\mathsf{\Delta }}_{r_f}$ we have

$$\Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega f_{\tau }^{\prime \prime }(\omega ;q)}{f_{\tau }^{\prime }(\omega ;q)}}\right)\right)>\sigma cos\gamma$$

if and only if $r_f$ is the unique root of

$${\displaystyle \frac{r{f}_{\tau }^{″}(r;q)}{f_{\tau }^{\prime }(r;q)}}+\left(1-\sigma \right)cos\gamma =0\mathrm{or}{\displaystyle \frac{r{J}_{\tau }^{″}(r;q)}{J_{\tau }^{\prime }(r;q)}}+\left({\displaystyle \frac{1}{\tau }}-1\right){\displaystyle \frac{r{J}_{\tau }^{\prime }(r;q)}{J_{\tau }(r;q)}}=\left(\sigma -1\right)cos\gamma ,$$

situated in $(0,j_{\tau ,1}^{\prime }\left(q\right)).$

For the other parts, note that the functions $g_{\tau }^{\left(s\right)}(\omega ;q)$ and $h_{\tau }^{\left(s\right)}(\omega ;q)$ for $s\in \left\{2,3\right\}$ belong to the Laguerre-Pólya class $\mathcal{LP}$, which is closed under differentiation, their derivatives $d{g}_{\tau }^{\left(s\right)}(\omega ;q)/d\omega$ and $d{h}_{\tau }{(\omega ;q)}^{\left(s\right)}/d\omega$ also belong to $\mathcal{LP}$ and the zeros are real for $\tau >-1.$ Thus, assuming ${\sigma }_{\tau ,k}\left(q\right)$ and ${\beta }_{\tau ,k}\left(q\right)$ are the k-th positive zeros of $\omega \mapsto d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega +(1-\tau )J_{\tau }^{\left(2\right)}(\omega ;q)$ and $\omega \mapsto d{J}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega +(2-\tau )J_{\tau }^{\left(2\right)}(\omega ;q)$$h_{\tau }^{\prime }$, while ${\gamma }_{\tau ,k}\left(q\right)$ and ${\delta }_{\tau ,k}\left(q\right)$ are the k-th positive zeros of $\omega \mapsto d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega +(1-\tau )J_{\tau }^{\left(3\right)}(\omega ;q)$ and $\omega \mapsto d{J}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega +(2-\tau )J_{\tau }^{\left(3\right)}(\omega ;q)$$h_{\tau }^{\prime }$, respectively, we have the following representations:

$$d{g}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega =\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\sigma }_{\tau ,k}^2\left(q\right)}}\right),d{h}_{\tau }^{\left(2\right)}(\omega ;q)/d\omega =\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\beta }_{\tau ,k}^2\left(q\right)}}\right),$$

and

$$d{g}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega =\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\gamma }_{\tau ,k}^2\left(q\right)}}\right),d{h}_{\tau }^{\left(3\right)}(\omega ;q)/d\omega =\prod _{k\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\delta }_{\tau ,k}^2\left(q\right)}}\right).$$

Similarly, we can prove parts $\left(ii\right)$ and $\left(iii\right).$ □

Remark 2. 

Taking $\gamma =0$ in Theorem 2 yields ([28] Theorem 2).

3. $\mathit{q}$-Bessel-Struve Functions

The Struve functions applications in other active research areas can be found in [45,46]. Recently, Oraby and Mansour [30] introduced the q-Struve-Bessel functions $H_{\tau }^{\left(s\right)}(\omega ;q^2),s\in \left\{2,3\right\},$ defined by

$$H_{\tau }^{\left(2\right)}(\omega ;q^2)=\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{q}^{2n^2+2n\tau +2n}}{{\mathsf{\Gamma }}_{q^2}\left(n+\frac{3}{2}\right){\mathsf{\Gamma }}_{q^2}\left(n+\tau +\frac{3}{2}\right)}}{\left({\displaystyle \frac{\omega }{1+q}}\right)}^{2n+\tau +1}$$

(15)

and

$$H_{\tau }^{\left(3\right)}(\omega ;q^2)=\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n{q}^{n^2+n}}{{\mathsf{\Gamma }}_{q^2}\left(n+\frac{3}{2}\right){\mathsf{\Gamma }}_{q^2}\left(n+\tau +\frac{3}{2}\right)}}{\left({\displaystyle \frac{\omega }{1+q}}\right)}^{2n+\tau +1},$$

(16)

where $\omega \in \mathbb{C}.$ we follow [47] for the definitions of q-shifted factorial, q-gamma function, the q-binomial coefficients, Jackson q-difference and q-integral operators and q-numbers.

The functions $H_{\tau }^{\left(s\right)}(\omega ;q^2),s\in \left\{2,3\right\}$ are q-form of the Struve function ([40] p. 328)

$$H_{\tau }\left(\omega \right)=\sum _{n\ge 0}{\displaystyle \frac{{(-1)}^n}{\mathsf{\Gamma }\left(n+\frac{3}{2}\right)\mathsf{\Gamma }\left(n+\tau +\frac{3}{2}\right)}}{\left({\displaystyle \frac{\omega }{2}}\right)}^{2n+\tau +1}.$$

That is,

$$\underset{q\to 1^{-}}{lim}{H}_{\tau }^{\left(2\right)}(\omega ;q^2)=\underset{q\to 1^{-}}{lim}{H}_{\tau }^{\left(3\right)}(\omega ;q^2)=H_{\tau }\left(\omega \right),s\in \left\{2,3\right\}.$$

One can see that the functions ${\omega }^{-\tau -1}{H}_{\tau }^{\left(s\right)}(\omega ;q^2),s\in \left\{2,3\right\}$ are entire functions of order zero and have infinitely many zeros. So Hadamard factorization theorem [37] says that

$${\omega }^{-\tau -1}{H}_{\tau }^{\left(2\right)}(\omega ;q^2)={\displaystyle \frac{{\left(1+q\right)}^{-\tau -1}}{{\mathsf{\Gamma }}_{q^2}\left(\frac{3}{2}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +\frac{3}{2}\right)}}\prod _{n\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)}}\right)$$

(17)

and

$${\omega }^{-\tau -1}{H}_{\tau }^{\left(3\right)}(\omega ;q^2)={\displaystyle \frac{{\left(1+q\right)}^{-\tau -1}}{{\mathsf{\Gamma }}_{q^2}\left(\frac{3}{2}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +\frac{3}{2}\right)}}\prod _{n\ge 1}\left(1-{\displaystyle \frac{{\omega }^2}{{\tilde{\theta }}_{\tau ,n}^2\left(q^2\right)}}\right)$$

(18)

where ${\theta }_{\tau ,n}\left(q^2\right)$ and ${\tilde{\theta }}_{\tau ,n}\left(q^2\right)$ denote the n-th positive zeros of the functions ${\omega }^{-\tau -1}{H}_{\tau }^{\left(2\right)}(\omega ;q^2)$ and ${\omega }^{-\tau -1}{H}_{\tau }^{\left(3\right)}(\omega ;q^2),$ respectively. Oraby and Mansour [30] proved that for $\tau >-{\displaystyle \frac{1}{2}}$ the functions ${\omega }^{-\tau -1}{H}_{\tau }^{\left(s\right)}(\omega ;q^2),s\in \left\{2,3\right\}$ have only real simple zeros. We also refer to [28,31,32].

Since $H_{\tau }^{\left(2\right)}(\omega ;q^2),H_{\tau }^{\left(3\right)}(\omega ;q^2)\notin \mathcal{A}$, we consider some normalizations as in [10,28]. For $\left|\tau \right|<{\displaystyle \frac{1}{2}},$ we associate $H_{\tau }^{\left(2\right)}(\omega ;q^2)$ with the normalized functions:

$$\begin{array}{ccc}\hfill k_{\tau }^{\left(2\right)}(\omega ;q^2)& =& {\left[{\left(1+q\right)}^{\tau }{\mathsf{\Gamma }}_{q^2}\left({\displaystyle \frac{1}{2}}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +{\displaystyle \frac{3}{2}}\right)H_{\tau }^{\left(2\right)}(\omega ;q^2)\right]}^{{\displaystyle \frac{1}{\tau +1}}},\left(\tau \ne -1\right),\hfill \\ \hfill l_{\tau }^{\left(2\right)}(\omega ;q^2)& =& {\left(1+q\right)}^{\tau }{\mathsf{\Gamma }}_{q^2}\left({\displaystyle \frac{1}{2}}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +{\displaystyle \frac{3}{2}}\right){\omega }^{-\tau }{H}_{\tau }^{\left(2\right)}(\omega ;q^2),\hfill \\ \hfill m_{\tau }^{\left(2\right)}(\omega ;q^2)& =& {\left(1+q\right)}^{\tau }{\mathsf{\Gamma }}_{q^2}\left({\displaystyle \frac{1}{2}}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +{\displaystyle \frac{3}{2}}\right){\omega }^{{\displaystyle \frac{1-\tau }{2}}}{H}_{\tau }^{\left(2\right)}(\sqrt{\omega };q^2).\hfill \end{array}$$

In a similar manner, we connect with $H_{\tau }^{\left(3\right)}(\omega ;q^2)$ the functions:

$$\begin{array}{ccc}\hfill k_{\tau }^{\left(3\right)}(\omega ;q^2)& =& {\left[{\left(1+q\right)}^{\tau }{\mathsf{\Gamma }}_{q^2}\left({\displaystyle \frac{1}{2}}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +{\displaystyle \frac{3}{2}}\right)H_{\tau }^{\left(3\right)}(\omega ;q^2)\right]}^{{\displaystyle \frac{1}{\tau +1}}},\left(\tau \ne -1\right),\hfill \\ \hfill l_{\tau }^{\left(3\right)}(\omega ;q^2)& =& {\left(1+q\right)}^{\tau }{\mathsf{\Gamma }}_{q^2}\left({\displaystyle \frac{1}{2}}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +{\displaystyle \frac{3}{2}}\right){\omega }^{-\tau }{H}_{\tau }^{\left(3\right)}(\omega ;q^2),\hfill \\ \hfill m_{\tau }^{\left(3\right)}(\omega ;q^2)& =& {\left(1+q\right)}^{\tau }{\mathsf{\Gamma }}_{q^2}\left({\displaystyle \frac{1}{2}}\right){\mathsf{\Gamma }}_{q^2}\left(\tau +{\displaystyle \frac{3}{2}}\right){\omega }^{{\displaystyle \frac{1-\tau }{2}}}{H}_{\tau }^{\left(3\right)}(\sqrt{\omega };q^2).\hfill \end{array}$$

Clearly, the functions $k_{\tau }^{\left(s\right)}(\omega ;q^2),l_{\tau }^{\left(s\right)}(\omega ;q^2)$ and $m_{\tau }^{\left(s\right)}(\omega ;q^2),s\in \left\{2,3\right\}$ belong to the class $\mathcal{A}$.

The following lemma is essential for the coming results.

Lemma 6 

([31]). Between any two consecutive the functions roots $H_{\tau }^{\left(s\right)}(\omega ;q^2),$ the function $d{H}_{\tau }^{\left(s\right)}(\omega ;q^2)/d\omega ,s\in \left\{2,3\right\}$ has precisely one zero when $\left|\tau \right|<{\displaystyle \frac{1}{2}}.$

Now we can determine the radii of $\gamma$-spirallike of order $\sigma$ of the above mentioned normalized functions.

Theorem 3. 

Let $\left|\tau \right|<{\displaystyle \frac{1}{2}},\sigma \in [0,1),$$\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ and $s\in \left\{2,3\right\}$. Then, the following statements are true:

(i)

$r_{sp}^{*}(\sigma ,\gamma ;k_{\tau })=x_{\tau ,\sigma ,\gamma ,2}$ is the lowest positive roots of the equation

$$r·d{H}_{\tau }^{\left(s\right)}(r;q^2)/dr-\left[1-\left(1-\sigma \right)cos\gamma \right]\left(\tau +1\right)H_{\tau }^{\left(s\right)}(r;q^2)=0.$$

(ii)

$r_{sp}^{*}(\sigma ,\gamma ;l_{\tau })=y_{\tau ,\sigma ,\gamma ,2}$ is the lowest positive roots of the equation

$$r·d{H}_{\tau }^{\left(s\right)}(r;q^2)/dr-\left[\tau +1-\left(1-\sigma \right)cos\gamma \right]H_{\tau }^{\left(s\right)}(r;q^2)=0.$$

(iii)

$r_{sp}^{*}(\sigma ,\gamma ;m_{\tau })=z_{\tau ,\sigma ,\gamma ,2}$ is the lowest positive roots of the equation

$$\sqrt{r}·d{H}_{\tau }^{\left(s\right)}(\sqrt{r};q^2)/dr-\left[\tau +1-2\left(1-\sigma \right)cos\gamma \right]H_{\tau }^{\left(s\right)}(\sqrt{r};q^2)=0.$$

Proof. 

The only difference between the proofs for the cases $s=2$ and $s=3$ is that the proofs use the different zeros ${\theta }_{\tau ,n}\left(q^2\right)$ and ${\tilde{\theta }}_{\tau ,n}\left(q^2\right)$. As a result, we will only provide the proof for the case $s=2$ in the following, and we will use the following notations for simplicity: $H_{\tau }(\omega ;q^2)=H_{\tau }^{\left(2\right)}(\omega ;q^2),$$k_{\tau }(\omega ;q^2)=k_{\tau }^{\left(2\right)}(\omega ;q^2),$$l_{\tau }(\omega ;q^2)=l_{\tau }^{\left(2\right)}(\omega ;q^2),$$m_{\tau }(\omega ;q^2)=m_{\tau }^{\left(2\right)}(\omega ;q^2)$ and $H_{\tau }^{\prime }(\omega ;q^2)=d{H}_{\tau }^{\left(2\right)}(\omega ;q^2)/d\omega .$

From (17), we have

$${\displaystyle \frac{\omega H_{\tau }^{\prime }(\omega ;q^2)}{H_{\tau }(\omega ;q^2)}}=\tau +1-\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}},$$

where ${\theta }_{\tau ,n}\left(q^2\right)$ stands to the n-th positive zero $H_{\tau }(\omega ;q^2).$ It follows that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\omega k_{\tau }^{\prime }(\omega ;q^2)}{k_{\tau }(\omega ;q^2)}}& =& {\displaystyle \frac{1}{\tau +1}}{\displaystyle \frac{\omega H_{\tau }^{\prime }(\omega ;q^2)}{H_{\tau }(\omega ;q^2)}}=1-{\displaystyle \frac{1}{\tau +1}}\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}},\hfill \\ \hfill {\displaystyle \frac{\omega l_{\tau }^{\prime }(\omega ;q^2)}{l_{\tau }(\omega ;q^2)}}& =& -\tau +{\displaystyle \frac{\omega H_{\tau }^{\prime }(\omega ;q^2)}{H_{\tau }(\omega ;q^2)}}=1-\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}},\hfill \\ \hfill {\displaystyle \frac{\omega m_{\tau }^{\prime }(\omega ;q^2)}{m_{\tau }(\omega ;q^2)}}& =& {\displaystyle \frac{1-\tau }{2}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\sqrt{\omega }{H}_{\tau }^{\prime }(\sqrt{\omega };q^2)}{H_{\tau }(\sqrt{\omega };q^2)}}=1-\sum _{n\ge 1}{\displaystyle \frac{\omega }{{\theta }_{\tau ,n}^2\left(q^2\right)-\omega }}.\hfill \end{array}$$

(19)

On the other hand, from the inequality (11)

$$\Re \left({\displaystyle \frac{{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}}\right)\le {\displaystyle \frac{{\left|\omega \right|}^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\left|\omega \right|}^2}}$$

holds for every $\left|\tau \right|<{\displaystyle \frac{1}{2}},n\in \mathbb{N}$ and $\left|\omega \right|<{\theta }_{\tau ,1}\left(q^2\right),$ which in turn, implies that for $\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right),$

$$\begin{array}{ccc}\hfill \Re \left(e^{-i\gamma }{\displaystyle \frac{\omega k_{\tau }^{\prime }(\omega ;q^2)}{k_{\tau }(\omega ;q^2)}}\right)& =& \Re \left(e^{-i\gamma }\right)-{\displaystyle \frac{1}{\tau +1}}\Re \left(e^{-i\gamma }\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}}\right)\hfill \\ & \ge & cos\gamma -{\displaystyle \frac{1}{\tau +1}}\sum _{n\ge 1}{\displaystyle \frac{2{\left|\omega \right|}^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\left|\omega \right|}^2}}={\displaystyle \frac{\left|\omega \right|k_{\tau }^{\prime }(\left|\omega \right|;q^2)}{k_{\tau }(\left|\omega \right|;q^2)}}+cos\gamma -1,\hfill \end{array}$$

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega l_{\tau }^{\prime }(\omega ;q^2)}{l_{\tau }(\omega ;q^2)}}\right)\ge {\displaystyle \frac{\left|\omega \right|l_{\tau }^{\prime }(\left|\omega \right|;q^2)}{l_{\tau }(\left|\omega \right|;q^2)}}+cos\gamma -1$$

and

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega m_{\tau }^{\prime }(\omega ;q^2)}{m_{\tau }(\omega ;q^2)}}\right)\ge {\displaystyle \frac{\left|\omega \right|m_{\tau }^{\prime }(\left|\omega \right|;q^2)}{m_{\tau }(\left|\omega \right|;q^2)}}+cos\gamma -1.$$

Then the minimum principle for harmonic functions [37] implies that

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega k_{\tau }^{\prime }(\omega ;q^2)}{k_{\tau }(\omega ;q^2)}}\right)>\sigma cos\gamma \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\left|\omega \right|<x_{\tau ,\sigma ,\gamma ,2},$$

where $x_{\tau ,\sigma ,\gamma ,2}$ is the lowest positive root of

$$r{H}_{\tau }^{\prime }(r;q^2)-\left[1-\left(1-\sigma \right)cos\gamma \right]\left(\tau +1\right)H_{\tau }(r;q^2)=0.$$

Similarly,

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega l_{\tau }^{\prime }(\omega ;q^2)}{l_{\tau }(\omega ;q^2)}}\right)>\sigma cos\gamma \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\left|\omega \right|<y_{\tau ,\sigma ,\gamma ,2},$$

where $y_{\tau ,\sigma ,\gamma ,2}$ is the lowest positive root of

$$r{H}_{\tau }^{\prime }(r;q^2)-\left[\tau +1-\left(1-\sigma \right)cos\gamma \right]H_{\tau }(r;q^2)=0$$

and

$$\Re \left(e^{-i\gamma }{\displaystyle \frac{\omega m_{\tau }^{\prime }(\omega ;q^2)}{m_{\tau }(\omega ;q^2)}}\right)>\sigma cos\gamma \mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\left|\omega \right|<{\omega }_{\tau ,\sigma ,\gamma ,2},$$

where $z_{\tau ,\sigma ,\gamma ,2}$ is the lowest positive root of

$$\sqrt{r}{H}_{\tau }^{\prime }(\sqrt{r};q^2)-\left[\tau +1-2\left(1-\sigma \right)cos\gamma \right]H_{\tau }(\sqrt{r};q^2)=0.$$

□

Remark 3. 

Taking $\gamma =0$ in Theorem 3 yields ([31] Theorems 3 and 4).

Our second result in this section concerns the radii of convex $\gamma$-spirallike of order $\sigma .$

Theorem 4. 

Let $\left|\tau \right|<{\displaystyle \frac{1}{2}},\sigma \in [0,1),$$\gamma \in \left(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ and $s\in \left\{2,3\right\}$. Then, the following statements hold.

(i)

The radius of convex γ-spirallike of order σ of $k_{\tau }^{\left(s\right)}(\omega ;q^2)$ is the lowest positive roots of the equation

$${\displaystyle \frac{r·d^2{H}_{\tau }^{\left(s\right)}(r;q^2)/d{r}^2}{d{H}_{\tau }^{\left(s\right)}(r;q^2)/dr}}+\left({\displaystyle \frac{1}{\tau +1}}-1\right){\displaystyle \frac{r·d{H}_{\tau }^{\left(s\right)}(r;q^2)/dr}{H_{\tau }(r;q^2)}}=\left(\sigma -1\right)cos\gamma .$$

(ii)

The radius of convex γ-spirallike of order σ of $l_{\tau }^{\left(s\right)}(\omega ;q^2)$ is the lowest positive roots of the equation

$${\displaystyle \frac{r^2·d^2{H}_{\tau }^{\left(s\right)}(r;q^2)/d{r}^2+\left(1-\tau \right)r·d{H}_{\tau }^{\left(s\right)}(r;q^2)/dr}{r·d{H}_{\tau }^{\left(s\right)}(r;q^2)/dr-\tau H_{\tau }^{\left(s\right)}(r;q^2)}}=1+\tau +\left(\sigma -1\right)cos\gamma .$$

(iii)

The radius of convex γ-spirallike of order σ of $m_{\tau }^{\left(s\right)}(\omega ;q^2)$ is the lowest positive roots of the equation

$${\displaystyle \frac{r·d^2{H}_{\tau }^{\left(s\right)}(\sqrt{r};q^2)/d{r}^2+\left(2-\tau \right)\sqrt{r}·d{H}_{\tau }^{\left(s\right)}(\sqrt{r};q^2)/dr}{\sqrt{r}·d{H}_{\tau }^{\left(s\right)}(\sqrt{r};q^2)/dr+\left(1-\tau \right)H_{\tau }^{\left(s\right)}(\sqrt{r};q^2)}}=1+\tau +2\left(\sigma -1\right)cos\gamma .$$

Proof. 

We only present the proof for the case s = 2, and the rest follows in parallel lines. For the sake of clarity, we will use the following notations in what follows: $H_{\tau }(\omega ;q^2)=H_{\tau }^{\left(2\right)}(\omega ;q^2),$$k_{\tau }(\omega ;q^2)=k_{\tau }^{\left(2\right)}(\omega ;q^2),$$l_{\tau }(\omega ;q^2)=l_{\tau }^{\left(2\right)}(\omega ;q^2),$$m_{\tau }(\omega ;q^2)=m_{\tau }^{\left(2\right)}(\omega ;q^2)$, $H_{\tau }^{\prime }(\omega ;q^2)=d{H}_{\tau }^{\left(2\right)}(\omega ;q^2)/d\omega$ and $H_{\tau }^{\prime \prime }(\omega ;q^2)=d^2{H}_{\tau }^{\left(2\right)}(\omega ;q^2)/d{\omega }^2.$

$\left(i\right)$ Let ${\theta }_{\tau ,n}\left(q^2\right)$ and ${\theta }_{\tau ,n}^{\prime }\left(q^2\right)$ be the positive roots of $H_{\tau }(\omega ;q^2)$ and $H_{\tau }^{\prime }(\omega ;q^2),$ respectively. In ([31] p. 76) the following equality was shown:

$$\begin{array}{ccc}\hfill 1+{\displaystyle \frac{\omega k_{\tau }^{\prime \prime }(\omega ;q^2)}{k_{\tau }^{\prime }(\omega ;q^2)}}& =& 1+{\displaystyle \frac{\omega H_{\tau }^{\prime \prime }(\omega ;q^2)}{H_{\tau }^{\prime }(\omega ;q^2)}}+\left({\displaystyle \frac{1}{\tau +1}}-1\right){\displaystyle \frac{\omega H_{\tau }^{\prime }(\omega ;q^2)}{H_{\tau }(\omega ;q^2)}}\hfill \\ & =& 1-\left({\displaystyle \frac{1}{\tau +1}}-1\right)\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}}-\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^{\prime 2}\left(q^2\right)-{\omega }^2}}.\hfill \end{array}$$

Now, suppose that $\tau \in \left[-{\displaystyle \frac{1}{2}},0\right].$ By using the inequality (11), for all $\omega \in {\mathsf{\Delta }}_{{\theta }_{\tau ,1}^{\prime }\left(q^2\right)},$ we obtain

$$\begin{array}{cc}\hfill & \Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega k_{\tau }^{\prime \prime }(\omega ;q^2)}{k_{\tau }^{\prime }(\omega ;q^2)}}\right)\right)\hfill \\ \hfill & =\Re \left(e^{-i\gamma }\right)-\Re \left(e^{-i\gamma }\left(\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^{\prime 2}\left(q^2\right)-{\omega }^2}}+\left({\displaystyle \frac{1}{\tau +1}}-1\right)\sum _{n\ge 1}{\displaystyle \frac{2{\omega }^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-{\omega }^2}}\right)\right)\hfill \\ \hfill & \ge cos\gamma -\left({\displaystyle \frac{1}{\tau +1}}-1\right)\sum _{n\ge 1}{\displaystyle \frac{2r^2}{{\theta }_{\tau ,n}^2\left(q^2\right)-r^2}}-\sum _{n\ge 1}{\displaystyle \frac{2r^2}{{\theta }_{\tau ,n}^{\prime 2}\left(q^2\right)-r^2}}\hfill \\ \hfill & \ge cos\gamma +{\displaystyle \frac{r{k}_{\tau }^{\prime \prime }(r;q^2)}{k_{\tau }^{\prime }(r;q^2)}},\hfill \end{array}$$

(20)

where $\left|\omega \right|=r.$ Moreover, if we use the inequality (14), then we obtain inequality (20) for $\tau \in \left[0,{\displaystyle \frac{1}{2}}\right].$ Here, we used that the zeros ${\theta }_{\tau ,n}\left(q^2\right)$ and ${\theta }_{\tau ,n}^{\prime }\left(q^2\right)$ interlace according to Lemma 6. The inequality (20) implies for $r\in \left(0,{\theta }_{\tau ,1}^{\prime }\left(q^2\right)\right)$

$$\underset{\omega \in {\mathsf{\Delta }}_r}{inf}\left\{\Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega k_{\tau }^{\prime \prime }(\omega ;q^2)}{k_{\tau }^{\prime }(\omega ;q^2)}}\right)-\sigma cos\gamma \right)\right\}=\left(1-\sigma \right)cos\gamma +{\displaystyle \frac{r{k}_{\tau }^{″}(r;q^2)}{k_{\tau }^{\prime }(r;q^2)}}.$$

Opposed to that, we define the function ${\mathsf{\Theta }}_{\tau }(r;q^2):\left(0,{\theta }_{\tau ,1}^{\prime }\left(q^2\right)\right)\to \mathbb{R},$

$${\mathsf{\Theta }}_{\tau }(r;q^2)=\left(1-\sigma \right)cos\gamma +{\displaystyle \frac{r{k}_{\tau }^{″}(r;q^2)}{k_{\tau }^{\prime }(r;q^2)}}$$

is strictly decreasing for $\left|\tau \right|<{\displaystyle \frac{1}{2}}$ and $r<\sqrt{{\theta }_{\tau ,n}\left(q^2\right){\theta }_{\tau ,n}^{\prime }\left(q^2\right)}.$ Since ${lim}_{r\searrow 0}{\mathsf{\Theta }}_{\tau }(r;q^2)=\left(1-\sigma \right)cos\gamma >0$ and ${lim}_{r\nearrow {\theta }_{\tau ,1}^{\prime }\left(q\right)}{\mathsf{\Theta }}_{\tau }(r;q^2)=-\infty ,$ then for $\omega \in {\mathsf{\Delta }}_{r_k},$ we have

$$\Re \left(e^{-i\gamma }\left(1+{\displaystyle \frac{\omega k_{\tau }^{\prime \prime }(\omega ;q^2)}{k_{\tau }^{\prime }(\omega ;q^2)}}\right)\right)>\sigma cos\gamma$$

if and only if $r_k$ is the equation’s lowest positive root

$${\displaystyle \frac{r{H}_{\tau }^{″}(r;q^2)}{H_{\tau }^{\prime }(r;q^2)}}+\left({\displaystyle \frac{1}{\tau +1}}-1\right){\displaystyle \frac{r{H}_{\tau }^{\prime }(r;q^2)}{H_{\tau }(r;q^2)}}=\left(\sigma -1\right)cos\gamma ,$$

situated in $(0,{\theta }_{\tau ,1}^{\prime }\left(q^2\right)).$ Likewise, we can show parts $\left(ii\right)$ and $\left(iii\right).$ □

Remark 4. 

Taking $\gamma =0$ in Theorem 4 yields ([31] Theorem 8 and 9).

4. Conclusions

In conclusion, we have investigated the radii of $\gamma$-spirallike and convex $\gamma$-spirallike of order $\sigma$ of normalized forms for the functions q-Bessel and q-Bessel-Struve in the unit disk of the complex plane. These results extend existing findings for classical Bessel and Struve functions to their q-generalizations. The characterization of entire functions within the Laguerre-Pólya class, facilitated by hyperbolic polynomials, has been pivotal in our proofs. Furthermore, leveraging the interlacing property of zeros between q-Bessel and q-Bessel-Struve functions and their derivatives has significantly contributed to establishing our main theorems. In the main results, when we take special values for $\gamma ,$ previously obtained results were found.

This work provides significant contributions to understanding the geometric properties of q-special functions. It also highlights the potential for future research to explore similar radius properties in detail for different functions.