Majorization Problem for q-General Family of Functions with Bounded Radius Rotations

Abstract

In this paper, we first prove the q-version of Schwarz Pick’s lemma. This result improved the one presented earlier in the literature without proof. Using this novel result, we study the majorization problem for the q-general class of functions with bounded radius rotations, which we introduce here. In addition, the coefficient bound for majorized functions related to this class is derived. Relaxing the majorized condition on this general family, we obtain the estimate of coefficient bounds associated with the class. Consequently, we present new results as corollaries and point out relevant connections between the main results obtained from the ones in the literature.

1. Introduction and Preliminaries

We consider $\mathcal{A}$ as the class of normalized analytic functions f in $\mho :=\{t\in \mathbb{C}:|t|<1\}$ that satisfy the condition $f\left(0\right)=f^{\prime }\left(0\right)-1=0$ and have the following series.

$$f\left(t\right)=t+\sum _{n=2}^{\infty }{a}_n{t}^n,t\in \mho .$$

(1)

The function $f\in \mathcal{A}$ is subordinate to function $g\in \mathcal{A}$ if there exists an analytic function $\mathsf{\omega }:\mho ⟶\mho$ with $\mathsf{\omega }\left(0\right)=0$, such that $f\left(t\right)=(g\circ \mathsf{\omega })\left(t\right)$. The family of all univalent functions in $\mathcal{A}$ is denoted by $\mathcal{S}$. The subclasses ${\mathcal{S}}^{\ast }$ and $\mathcal{C}$ of $\mathcal{S}$ represent the usual families of starlike and convex functions in ℧, respectively. These classes were first comprehensively classified and studied by Shanmugam [1] using the properties of convolution and subordination. Later, Ma and Minda [2] established their coefficient-related results. They considered the analytic function $\varphi$ with the following properties:

(a)

$\varphi \in \mathcal{S}$ with $\mathrm{Re}\varphi \left(t\right)>0$,

(b)

$\varphi$ is starlike with respect to $\varphi \left(0\right)=1$,

(c)

$\varphi (\mho )$ is symmetric about the real axis,

(d)

${\varphi }^{\prime }\left(0\right)>0$.

Then, they defined the following two families:

$${\mathcal{S}}^{\ast }\left(\varphi \right):=\left\{f\in \mathcal{A}:{\displaystyle \frac{t{f}^{\prime }\left(t\right)}{f\left(t\right)}}\prec \varphi \left(t\right)\right\}$$

(2)

and

$$\mathcal{C}\left(\varphi \right):=\left\{f\in \mathcal{A}:{\displaystyle \frac{{\left(t{f}^{\prime }\left(t\right)\right)}^{\prime }}{f^{\prime }\left(t\right)}}\prec \varphi \left(t\right)\right\}.$$

(3)

These classes laid a solid foundation for the emergence of various families of their subclasses. The table below illustrates some of the subfamilies of ${\mathcal{S}}^{\ast }\left(\varphi \right)$ related to the present investigation.

S/N	${\mathcal{S}}^{\ast }\left(\varphi \right)$	$\varphi$
(i)	${\mathcal{S}}_{\mathcal{E}}^{\ast }$	${\varphi }_1\left(t\right)={\mathrm{e}}^t$ [3]
(ii)	${\mathcal{S}}_L^{\ast }\left(s\right)$	${\varphi }_2\left(t\right)={(1+st)}^2,s\in (0,1/\sqrt{2}]$ [4]
(iii)	${\mathcal{S}}_C^{\ast }$	${\varphi }_3\left(t\right)=t+\sqrt{1+t^2}$ [5]
(iv)	${\mathcal{S}}^{\ast }(A,B)$	${\varphi }_4\left(t\right)=(1+At)/(1+Bt),-1\le B<A\le 1$ [6]
(v)	${\mathcal{S}}_{\mathsf{\alpha }}^{\ast }$	${\varphi }_5\left(t\right)=(1+(1-2\mathsf{\alpha })t)/(1-t),\mathsf{\alpha }\in [0,1)$ [7]
(vi)	${\mathcal{S}}_{L_q}^{\ast }\left(s\right)$	${\varphi }_6\left(t\right)={\left({\displaystyle \frac{2(1+st)}{2+s(1-q)t}}\right)}^2,s\in (0,1/\sqrt{2}]$ [8,9] .

The q-calculus (i.e., quantum calculus) is a generalization of classical calculus without the concept of limit. Instead, it uses the quantity q as a parameter to extend many classical results, methods, and subclasses of $\mathcal{S}$. It was first introduced in geometric function theory by Ismail et al. [10]. Consequently, many articles in this direction appeared in the literature. For example, one may see [8,9,11,12] and the references. Mohammed and Adegani [12] extended the general Ma-Minda class to that of its q-analogue and obtained the radius of majorization associated with the class.

Definition 1.

Let $0<q<1$. Then the q-number ${\left[n\right]}_q$ is given by

$${\left[n\right]}_q=\left\{\begin{array}{c}{\displaystyle \frac{1-q^n}{1-q}},n\in \mathbb{C},\hfill \\ {\displaystyle \sum _{\iota =0}^{n-1}}{q}^{\iota }=1+q+q^2+\cdots q^{n-1},n\in \mathbb{N},\hfill \\ n,\mathrm{as}q\to 1^{-},\hfill \end{array}\right.$$

while the q-factorial and q-derivative of a complex valued function f in $\mathcal{A}$ are respectively given by

$${\left[n\right]}_q!=\left\{\begin{array}{cc}1\mathrm{if}n=0\hfill & \hfill \\ {\left[n\right]}_q·{[n-1]}_q·{[n-2]}_q·{[n-3]}_q\cdots 2·1\mathrm{if}n\in \mathbb{N}\hfill \end{array}\right.$$

$$D_{q}f\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{f\left(qz\right)-f\left(t\right)}{(q-1)t}},t\ne 0\hfill \\ f^{\prime }\left(0\right),t=0,\hfill \\ f^{\prime }\left(t\right),\mathrm{as}q\to 1^{-}.\hfill \end{array}\right.$$

From the above explanation, it is easy to see that for $f\left(t\right)$ given by (1),

$$D_{q}f\left(t\right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{t}^n.$$

Let $f,g\in \mathcal{A}$, we have the following rules for q-difference operator $D_q$.

(i)

$D_q\left(f\left(t\right)g\left(t\right)\right)=f\left(qt\right)D_{q}g\left(t\right)+g\left(t\right)D_{q}f\left(t\right)$;

(ii)

$D_q(\sigma f\left(t\right)\pm \delta g\left(t\right))=\sigma D_{q}f\left(t\right)\pm \delta D_{q}g\left(t\right)$, for $\sigma ,\delta \in \mathbb{C}\setminus \left\{0\right\}$.

A natural generalization of the class ${\mathcal{S}}^{\ast }$ is the class ${\mathcal{S}}_{\mathsf{\kappa }}^{\ast }$ of functions with bounded radius rotation whose investigation started with the work of Leowner [13] and subsequently developed by Paatero [14,15]. According to Pinchuk [16], a function $f\in {\mathcal{S}}_{\mathsf{\kappa }}^{\ast }$ if and only if for $\mathsf{\kappa }\ge 2$, it has representation:

$${\mathcal{S}}_f\left(t\right)=\left({\displaystyle \frac{\mathsf{\kappa }+2}{4}}\right)p_1\left(t\right)-\left({\displaystyle \frac{\mathsf{\kappa }-2}{4}}\right)p_2\left(t\right),$$

where $p_j\left(t\right)\prec (1+t)/(1-t)(j=1,2)$.

Recently, this class received attention in the direction of Ma-Minda class. Afis and Noor [17] and Jabeen and Saliu [18] introduced the classes ${\mathcal{S}}_{C_{\mathsf{\kappa }}}^{\ast }$ and ${\mathcal{S}}_{L_{\mathsf{\kappa }}}^{\ast }\left(s\right)$ of bounded radius rotation related with crescent and limaçon domains. They examined the inclusion properties, radius problems, coefficient inequalities, and other associated geometrical properties. For more information on the study of ${\mathcal{S}}_{\mathsf{\kappa }}^{\ast }$ and its subfamilies, one may see “A survey on functions of bounded boundary and bounded radius rotation” by Noor [19].

Let $f,g\in \mathcal{A}$. Then f is majorized by g or g majorized f (written as $f\left(t\right)\ll g\left(t\right)$) in ℧ if $\left|f\right(t\left)\right|\le \left|g\right(t\left)\right|$ for all $t\in \mho$. The idea of this notion emanated from the work of Macgregor [20], where he showed that if $f\left(t\right)\ll g\left(t\right)$ in ℧, then there exists an analytic function $\phi$ with $\left|\phi \right(t\left)\right|\le 1$ in ℧ such that

$$f\left(t\right)=\phi \left(t\right)g\left(t\right),t\in \mho .$$

This article opened the door to studying the majorization problems for various classes of univalent functions (see [12,21,22,23] and the references cited therein). Cho et al. [21] examined the majorization property for the class ${\mathcal{S}}^{\ast }\left(\varphi \right)$, which was later generalized by Mohammed and Adegani [12] and Adegani et al. [23].

Motivated by the recent work [12,17,18,21,24], we introduce and investigate the majorization problem for the general class ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$ of functions of bounded radius rotation. In addition, we obtain the coefficient inequality for majorized functions associated with this class and illustrate some particular cases of our results. Furthermore, relaxing the majorized condition on this general family, we obtain the estimate of coefficient bounds associated with the class.

Definition 2.

Let $f\in \mathcal{A}$. Then $f\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$ if and only for $\mathsf{\kappa }\ge 2$,

$${\displaystyle \frac{t{D}_{q}f\left(t\right)}{f\left(t\right)}}=\left({\displaystyle \frac{\mathsf{\kappa }+2}{4}}\right)p_1\left(t\right)-\left({\displaystyle \frac{\mathsf{\kappa }-2}{4}}\right)p_2\left(t\right),$$

where $p_j\left(t\right)\prec \varphi \left(t\right)(j=1,2)$.

As a special case, we have the following:

(a)

As $q\to 1^{-}$ the class ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$ reduces to the one introduced and studied by Jabeen et al. [24].

(b)

For $\mathsf{\kappa }=2$, we have the class introduced and studied by Mohammad and Adegani [12].

(c)

For $\mathsf{\kappa }=2$ and $q\to 1^{-}$, we obtain the general family of Ma-Minda [2].

Remark 1.

When we replace $\varphi \left(t\right)$ with Ma and Minda functions, we obtain the corresponding subclasses of q-bounded radius rotations of Ma and Minda type (see [25]).

Remark 2.

When we replace $\varphi \left(t\right)$ with Ma and Minda functions and suppose $q\to 1^{-}$, we obtain the corresponding subclasses of bounded radius rotations of Ma and Minda type (see [17,18]).

Remark 3.

When we replace $\varphi \left(t\right)$ with Ma and Minda functions and suppose $\mathsf{\kappa }=2$, we obtain q-starlike class of Ma and Minda type (see [8,9,26]).

Remark 4.

When we replace $\varphi \left(t\right)$ with Ma and Minda functions, $\mathsf{\kappa }=2$ and $q\to 1^{-}$, we obtain the starlike class of Ma and Minda type (see [4,5,6]).

Next, we need the q-analogue of Schwarz Pick’s Lemma to prove our main result, which can be proved by using Nehari’s techniques at [27] (p. 168). This new lemma improves the earlier results by Adegani et al. [28] and by Vijaya et al. [29] without proof.

Lemma 1

($\mathit{q}$-Schwarz Pick’s Lemma). If ω is analytic in ℧ and $|\omega (t)|<1$, then

$$|D_q\omega \left(t\right)|\le {\displaystyle \frac{1-{\left|\omega \left(t\right)\right|}^2}{\left(1-q\left|t\right|\right)\left(1+\left|t\right|\right)}},t\in \mho .$$

Proof. 

Since $\omega$ is bounded in ℧, so is

$$g\left(t\right)={\displaystyle \frac{\omega \left(t\right)-\omega \left(\zeta \right)}{1-\overline{\omega \left(\zeta \right)}\omega \left(t\right)}},\left|\zeta \right|<1.$$

Observe that $g\left(\zeta \right)=0$. Consider

$$h\left(t\right)={\displaystyle \frac{g\left(t\right)(1-\overline{\zeta }t)}{t-\zeta }}.$$

(4)

Then, h is analytic at $t=\zeta ,\underset{\left|t\right|\to 1}{lim\; sup}\left|g\left(t\right)\right|\le 1$, and $\left|{\displaystyle \frac{t-\zeta }{1-t\overline{\zeta }}}\right|=1$ for $\left|t\right|=1$. Hence, by the maximum modulus principle, $\left|h\right(t\left)\right|\le 1$ for all $t\in \mho$. Now, setting $\zeta =qt$ in (4), we have

$$\begin{array}{cc}\hfill h\left(t\right)& =\left({\displaystyle \frac{\omega \left(t\right)-\omega \left(qt\right)}{t-qt}}\right)\left({\displaystyle \frac{1-q\overline{t}t}{1-\overline{\omega \left(qt\right)}\omega \left(t\right)}}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{1-{q\left|t\right|}^2}{1-\overline{\omega \left(qt\right)}\omega \left(t\right)}}\right)D_q\omega \left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{\left(1-{q\left|t\right|}^2\right)D_q\omega \left(t\right)}{1-\left(\overline{\omega \left(t\right)}-(1-q)\overline{t{D}_q\omega \left(t\right)}\right)\omega \left(t\right)}}\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \left|h\right(t\left)\right|& =\left|{\displaystyle \frac{\left(1-{q\left|t\right|}^2\right)D_q\omega \left(t\right)}{1-\left(\overline{\omega \left(t\right)}-(1-q)\overline{t{D}_q\omega \left(t\right)}\right)\omega \left(t\right)}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left(1-{q\left|t\right|}^2\right)\left|D_q\omega \left(t\right)\right|}{1-{\left|\omega \left(t\right)\right|}^2-(1-q)\left|t\right||D_q\omega \left(t\right)\left|\right|\omega \left(t\right)|}}\hfill \\ \hfill & <{\displaystyle \frac{\left(1-{q\left|t\right|}^2\right)\left|D_q\omega \left(t\right)\right|}{1-{\left|\omega \left(t\right)\right|}^2-(1-q)\left|t\right||D_q\omega \left(t\right)|}}.\hfill \end{array}$$

We see that $\left|h\right(t\left)\right|\le 1$ if

$${\displaystyle \frac{\left(1-{q\left|t\right|}^2\right)\left|D_q\omega \left(t\right)\right|}{1-{\left|\omega \left(t\right)\right|}^2-(1-q)\left|t\right||D_q\omega \left(t\right)|}}\le 1,$$

which holds if

$$\left(1-{q\left|t\right|}^2\right)|D_q{\omega \left(t\right)|\le 1-|\omega \left(t\right)|}^2-(1-q)\left|t\right||D_q\omega \left(t\right)|.$$

That is

$$\left(1-q\left|t\right|\right)(1+|t\left|\right)|D_q{\omega \left(t\right)|1-|\omega \left(t\right)|}^2.$$

Hence,

$$|D_q\omega \left(t\right)|\le {\displaystyle \frac{1-{\left|\omega \left(t\right)\right|}^2}{\left(1-q\left|t\right|\right)(1+|t\left|\right)}}.$$

□

Remark 5.

Lemma 1 reduces to the classic Schwarz Pick’s lemma as $q\to 1^{-}$.

Lemma 2

([30] (Proposition 5.4)). If ϕ is convex univalent, has non-negative Taylor coefficients about the origin, and $\mathrm{Re}\left(\varphi \right(t\left)\right)>0$ in ℧, then

$$\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|=\varphi (-r)\le \left|\varphi \left(t\right)\right|\le \varphi \left(r\right)=\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right|.$$

2. Main Results

Theorem 1.

If $f\in \mathcal{A}$ is majorized by $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$, then $D_{q}g\left(t\right)$ majorized $D_{q}f\left(t\right)$ in the disc $\left|t\right|<r_{\mathsf{\kappa }}\left(q\right)$, where $r_{\mathsf{\kappa }}\left(q\right)$ is the smallest positive root of the equation:

$$\left((1-qr)(1+r)-2r(1-q)\right)\left|(\mathsf{\kappa }+2)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-(\mathsf{\kappa }-2)\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right|\right|-8r=0.$$

Proof. 

From the antecedent, there exists an analytic function $\phi$ with $\left|\phi \right(t\left)\right|\le 1,t\in \mho$ such that $f\left(t\right)=\phi \left(t\right)g\left(t\right)$. Therefore,

$$\begin{array}{cc}\hfill D_{q}f\left(t\right)& =\phi \left(qt\right)D_{q}g\left(t\right)+D_q\phi \left(t\right)g\left(t\right)\hfill \\ \hfill & =\left(\phi \left(t\right)-(1-q)t{D}_q\phi \left(t\right)\right)D_{q}g\left(t\right)+g\left(t\right)D_q\phi \left(t\right)\hfill \\ \hfill & =\left(\phi \left(t\right)-(1-q)t{D}_q\phi \left(t\right)+{\displaystyle \frac{g\left(t\right)D_q\phi \left(t\right)}{D_{q}g\left(t\right)}}\right)D_{q}g\left(t\right)\hfill \\ \hfill & =\left(\phi \left(t\right)+\left({\displaystyle \frac{g\left(t\right)}{D_{q}g\left(t\right)}}-(1-q)t\right)D_q\phi \left(t\right)\right)D_{q}g\left(t\right).\hfill \end{array}$$

(5)

Since $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\phi \right)$, then

$$\begin{array}{cc}\hfill {\displaystyle \frac{t{D}_{q}g\left(t\right)}{g\left(t\right)}}& =\left({\displaystyle \frac{\mathsf{\kappa }+2}{4}}\right)p_1\left(t\right)-\left({\displaystyle \frac{\mathsf{\kappa }-2}{4}}\right)p_2\left(t\right),p_j\left(t\right)\prec \varphi \left(t\right)(j=1,2)\hfill \\ \hfill & =\left({\displaystyle \frac{\mathsf{\kappa }+2}{4}}\right)\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)-\left({\displaystyle \frac{\mathsf{\kappa }-2}{4}}\right)\varphi \left({\mathsf{\omega }}_2\left(t\right)\right),\hfill \end{array}$$

(6)

where ${\mathsf{\omega }}_j$ ($j=1,2$) is analytic in ℧ with ${\mathsf{\omega }}_j\left(0\right)=0$ such that $|{\mathsf{\omega }}_j\left(t\right)|=r<1$. Since $\mathrm{Re}\left(\varphi \left({\mathsf{\omega }}_j\left(t\right)\right)\right)>0(j=1,2)$ in ℧, we can rewrite (6) as

$${\displaystyle \frac{g\left(t\right)}{D_{q}g\left(t\right)}}={\displaystyle \frac{t}{\left(\frac{\mathsf{\kappa }+2}{4}\right)\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)-\left(\frac{\mathsf{\kappa }-2}{4}\right)\varphi \left({\mathsf{\omega }}_2\left(t\right)\right)}}.$$

(7)

From the minimum and maximum modulus principle, we have that

$$\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|\le \underset{|{\mathsf{\omega }}_1\left(t\right)|=r}{min}|\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)|=\underset{|{\mathsf{\omega }}_1\left(t\right)|\le r}{min}\left|\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)\right|$$

and

$$\underset{|{\mathsf{\omega }}_2\left(t\right)|\le r}{max}|\varphi \left({\mathsf{\omega }}_2\left(t\right)\right)|=\underset{|{\mathsf{\omega }}_2\left(t\right)|=r}{max}|\varphi \left({\mathsf{\omega }}_2\left(t\right)\right)|\le \underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right|.$$

Using these inequalities in (7), we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{g\left(t\right)}{D_{q}g\left(t\right)}}\right|& \le {\displaystyle \frac{r}{|\left(\frac{\mathsf{\kappa }+2}{4}\right)\left|\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)\right|-\left(\frac{\mathsf{\kappa }-2}{4}\right)\left|\varphi \left({\mathsf{\omega }}_2\left(t\right)\right)\right||}}\hfill \\ \hfill & \le {\displaystyle \frac{4r}{|\left(\mathsf{\kappa }+2\right)\underset{|{\mathsf{\omega }}_1\left(t\right)|=r}{min}\left|\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)\right|-\left(\mathsf{\kappa }-2\right)\underset{|{\mathsf{\omega }}_2\left(t\right)|=r}{max}\left|\varphi \left({\mathsf{\omega }}_2\left(t\right)\right)\right||}}\hfill \\ \hfill & \le {\displaystyle \frac{4r}{|\left(\mathsf{\kappa }+2\right)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-\left(\mathsf{\kappa }-2\right)\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right||}}.\hfill \end{array}$$

(8)

In view of (5) and (8), and using Lemma 1 with $\left|\phi \right(t\left)\right|=\rho \le 1$, we obtain

$$\begin{array}{c}|D_{q}f\left(t\right)|\hfill \\ \le \left(\left|\phi \left(t\right)\right|+\left(\left|{\displaystyle \frac{g\left(t\right)}{D_{q}g\left(t\right)}}\right|+(1-q)r\right)\left|D_q\phi \left(t\right)\right|\right)\left|D_{q}g\left(t\right)\right|\hfill \\ \le \left[\left|\phi \left(t\right)\right|+\left({\displaystyle \frac{4r}{|\left(\mathsf{\kappa }+2\right)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-\left(\mathsf{\kappa }-2\right)\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right||}}+(1-q)r\right)\left({\displaystyle \frac{1-{\left|\phi \left(t\right)\right|}^2}{(1-qr)(1+r)}}\right)\right]\left|D_{q}g\left(t\right)\right|\hfill \\ \le \left[\rho +\left({\displaystyle \frac{4r(1-{\rho }^2)}{(1-qr)(1+r)|\left(\mathsf{\kappa }+2\right)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-\left(\mathsf{\kappa }-2\right)\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right||}}+{\displaystyle \frac{r(1-q)(1-{\rho }^2)}{(1-qr)(1+r)}}\right)\right]\left|D_{q}g\left(t\right)\right|\hfill \\ :={\mathcal{V}}_{\mathsf{\kappa },q}(r,\rho ),r\in (0,1),\rho \in [0,1].\hfill \end{array}$$

To find $r_{\mathsf{\kappa }}\left(q\right)$, we choose $r_{\mathsf{\kappa }}\left(q\right)=max\left\{r\in (0,1):{\mathcal{V}}_{\mathsf{\kappa },q}(r,\rho )\le 1,\rho \in [0,1]\right\}$. But ${\mathcal{V}}_{\mathsf{\kappa },q}(r,\rho )\le 1$ if and only if

$$\begin{array}{cc}\hfill 0& \le \left((1-qr)(1+r)-r(1-q)(1+\rho )\right)|\left(\mathsf{\kappa }+2\right)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-\left(\mathsf{\kappa }-2\right)\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right||\hfill \\ \hfill & -4r(1+\rho ):={\mathcal{W}}_{\mathsf{\kappa },q}(r,\rho ).\hfill \end{array}$$

We see that ${\mathcal{W}}_{\mathsf{\kappa },q}(r,\rho )$ is decreasing on $0\le \rho \le 1$. Therefore, $min{\mathcal{W}}_{\mathsf{\kappa },q}(r,\rho )={\mathcal{W}}_{\mathsf{\kappa },q}(r,1):={\mathcal{W}}_{\mathsf{\kappa },q}\left(r\right)$, where

$${\mathcal{W}}_{\mathsf{\kappa },q}\left(r\right)=\left((1-qr)(1+r)-2r(1-q)\right)|\left(\mathsf{\kappa }+2\right)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-\left(\mathsf{\kappa }-2\right)\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right||-8r.$$

We observe that ${\mathcal{W}}_{\mathsf{\kappa },q}\left(0\right)>0$ and ${\mathcal{W}}_{\mathsf{\kappa },q}\left(1\right)<0$. Therefore, there exists $r=r_{\mathsf{\kappa }}\left(q\right)$ which satisfies (1). Hence, ${\mathcal{W}}_{\mathsf{\kappa },q}\left(r\right)\ge 0$ for all r in the disc $\left|t\right|<r_{\mathsf{\kappa }}\left(q\right)$. □

It is worthy of note that for $\mathsf{\kappa }=2$, we have:

Remark 6.

If $f\in \mathcal{A}$ is majorized by $g\in {\mathcal{S}}_q^{\ast }\left(\varphi \right)$, then $D_{q}g\left(t\right)$ majorized $D_{q}f\left(t\right)$ in the disc $\left|t\right|<r\left(q\right)$, where $r\left(q\right)$ is the smallest positive root of the equation:

$$\left((1-qr)(1+r)-2r(1-q)\right)\underset{\left|t\right|=r}{min}\left|\varphi \left(t\right)\right|-2r=0.$$

This result is an improvement to the one obtained by Mohammad and Adegani [12] (Theorem 2.1, p. 3). In addition, as $q\to 1^{-}$, Theorem 1 reduces to Theorem 2 in [21] and becomes various corollaries obtained for respective Ma and Minda functions. Now, for $\mathsf{\kappa }>2$, we obtain the following results.

Corollary 1.

Let $f\in \mathcal{A}$ and $f\left(t\right)\ll g\left(t\right)$ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_5\right):={\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\mathsf{\alpha }\right)$. Then $D_{q}f\left(t\right)\ll D_{q}g\left(t\right)$ in the disc $\left|t\right|<r_{\mathsf{\kappa },q}\left(\mathsf{\alpha }\right)$, where $r_{\mathsf{\kappa },q}\left(\mathsf{\alpha }\right)$ is the smallest positive root of the equation:

$$r_{\mathsf{\kappa },q}\left(\mathsf{\alpha }\right)=\left((1-qr)(1+r)-2r(1-q)\right)|(1-2\mathsf{\alpha })r^2-\mathsf{\kappa }(1-\mathsf{\alpha })r+1|-2r(1-r^2)=0$$

Proof. 

Since ${\varphi }_5\left(t\right)$ is convex in ℧, then by Lemma 2 and Theorem 1, we have the required result. □

We have the following consequence for $\mathsf{\alpha }=0$ in Corollary 1.

Corollary 2.

Let $f\in \mathcal{A}$ and $f\left(t\right)\ll g\left(t\right)$ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_5\right):={\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }$. Then $D_{q}f\left(t\right)\ll D_{q}g\left(t\right)$ in the disc $\left|t\right|<r_{\mathsf{\kappa },q}$, and $\left|t\right|<r_{\mathsf{\kappa },q}$ is the smallest positive root of the equation:

$$\left((1-qr)(1+r)-2r(1-q)\right)|r^2-\mathsf{\kappa }r+1|-2r(1-r^2)=0.$$

Corollary 3.

Let $f\in \mathcal{A}$ and $f\left(t\right)\ll g\left(t\right)$ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_2\right):={\mathcal{S}}_{L_{\mathsf{\kappa },q}}^{\ast }\left(s\right)$. Then $D_{q}g\left(t\right)$ majorized $D_{q}f\left(t\right)$ in the disc $\left|t\right|<r_{\mathsf{\kappa },q}\left(s\right)$, where $r_{\mathsf{\kappa },q}\left(s\right)$ is the smallest positive root of the equation

$$\left((1-qr)(1+r)-2r(1-q)\right)|s^2{r}^2-\mathsf{\kappa }sr+1|-2r=0.$$

Proof. 

The method of proof is the same as the proof of Corollary 1. □

Corollary 4.

Let $f\in \mathcal{A}$ and $f\left(t\right)\ll g\left(t\right)$ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_1\right):={\mathcal{S}}_{{\mathrm{e}}_{\mathsf{\kappa },q}}^{\ast }$. Then $D_{q}g\left(t\right)$ majorized $D_{q}f\left(t\right)$ in the disc $\left|t\right|<r_{\mathsf{\kappa },q}\left(\mathrm{e}\right)$, where $r_{\mathsf{\kappa },q}\left(\mathrm{e}\right)$ is the smallest roots of the equation

$$\left((1-qr)(1+r)-2r(1-q)\right)\left|2cosh\left(r\right)-\mathsf{\kappa }sinh\left(r\right)\right|-4r=0.$$

Remark 7.

As $q\to 1^{-}$, the Corollaries 1–4 reduce to the corresponding Corollaries 2.2–2.5 in [24].

Theorem 2.

Let $\varphi \left(t\right)=1+B_{1}t+B_2{t}^2+B_3{t}^3+\cdots$ be convex in ℧ and $f\in \mathcal{A}$ have the form (1). If $f\left(t\right)\ll g\left(t\right)$ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$, then

$$|a_n|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left({\displaystyle \frac{2q{[j-2]}_q+\mathsf{\kappa }\left|B_1\right|}{2q^{n-1}{[n-1]}_q!}}\right)\right].$$

Proof. 

Let

$${\displaystyle \frac{t{D}_{q}f\left(t\right)}{f\left(t\right)}}=p\left(t\right)=1+\sum _{n=1}^{\infty }{d}_n{t}^n.$$

It was shown in [24] that

$$|d_n|\le {\displaystyle \frac{\mathsf{\kappa }}{2}}\left|B_1\right|.$$

(9)

Now, carefully following the procedures of the proof in [24] (Theorem 2.6) and using (9), we obtain the required result. □

Remark 8.

(i) As $q\to 1^{-}$, Theorem 2 becomes Theorem 2.6 in [24].

(ii) For $\mathsf{\kappa }=2$ and when $q\to 1^{-}$, Theorem 2 reduces to the result of Cho et al. [21] (Theorem 3, p. 7).

Corollary 5.

Let $\varphi \left(t\right)=1+B_{1}t+B_2{t}^2+B_3{t}^3+\cdots$ be convex in ℧ and $f\in \mathcal{A}$ have the form (1). If $f\left(t\right)\ll g\left(t\right)$ with $g\in {\mathcal{S}}_q^{\ast }\left(\varphi \right)$, then

$$|a_n|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left({\displaystyle \frac{q{[j-2]}_q+\left|B_1\right|}{q^{n-1}{[n-1]}_q!}}\right)\right].$$

Corollary 6.

Let $f\in \mathcal{A}$ and suppose that $f\left(t\right)\ll g\left(t\right)$ in ℧ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_4\right):={\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }(A,B)$. Then

$$|a_n|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left({\displaystyle \frac{2q{[j-2]}_q+\mathsf{\kappa }(A-B)}{2q^{n-1}{[n-1]}_q!}}\right)\right].$$

Corollary 7.

Let $f\in \mathcal{A}$ and suppose $f\left(t\right)\ll g\left(t\right)$ in ${\mho }_{{\displaystyle \frac{\sqrt{2}}{2}}}:=\left\{z\in \mathbb{C}:\left|t\right|<\sqrt{2}/2\right\}$ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_3\right):={\mathcal{S}}_{C_{\mathsf{\kappa },q}}^{\ast }$. Then

$$|a_n|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left({\displaystyle \frac{2q{[j-2]}_q+\mathsf{\kappa }}{2q^{n-1}{[n-1]}_q!}}\right)\right].$$

Corollary 8.

Let $f\in \mathcal{A}$ and suppose that $f\left(t\right)\ll g\left(t\right)$ in ℧ with $g\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left({\varphi }_2\right):={\mathcal{S}}_{L_{\mathsf{\kappa },q}}^{\ast }$. Then for $0<s\le 1/2$,

$$|a_n|\le 1+\sum _{m=2}^n\left[\prod _{j=2}^m\left({\displaystyle \frac{q{[j-2]}_q+\mathsf{\kappa }s}{q^{n-1}{[n-1]}_q!}}\right)\right].$$

We note that the proof of Corollaries 2.10–2.113 follows the same method of proofs of Corollaries 2.7–2.9 in [24].

Remark 9.

As $q\to 1^{-}$, Corollaries 2.10–2.13 reduce to Corollaries 2.7–2.9 in [24].

In the next result, we relax the majorization condition and obtain the coefficient inequality associated with the class ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$.

Theorem 3.

If $f\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$, then

$$\sum _{n=2}^{\infty }\left(4{\left[n\right]}_q^2-{\mathsf{\kappa }}^2{\left(\underset{\left|t\right|=1}{max}\left|\varphi \left(t\right)\right|\right)}^2\right){\left|a_n\right|}^2\le {\mathsf{\kappa }}^2{\left(\underset{\left|t\right|=1}{max}\left|\varphi \left(t\right)\right|\right)}^2-4.$$

Proof. 

Since $f\in {\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$, then

$$\begin{array}{cc}\hfill {\displaystyle \frac{t{D}_{q}f\left(t\right)}{f\left(t\right)}}& =\left({\displaystyle \frac{\mathsf{\kappa }+2}{4}}\right)p_1\left(t\right)-\left({\displaystyle \frac{\mathsf{\kappa }-2}{4}}\right)p_2\left(t\right),p_j\left(t\right)\prec \varphi \left(t\right)(j=1,2)\hfill \\ \hfill & =\left({\displaystyle \frac{\mathsf{\kappa }+2}{4}}\right)\varphi \left({\mathsf{\omega }}_1\left(t\right)\right)-\left({\displaystyle \frac{\mathsf{\kappa }-2}{4}}\right)\varphi \left({\mathsf{\omega }}_2\left(t\right)\right),\hfill \end{array}$$

where ${\mathsf{\omega }}_j$ ($j=1,2$) is analytic in ℧ with ${\mathsf{\omega }}_j\left(0\right)=0$ such that $|{\mathsf{\omega }}_j\left(t\right)|=r<1$. Thus,

$$f\left(t\right)={\displaystyle \frac{4t{D}_{q}f\left(t\right)}{(\mathsf{\kappa }+2)\varphi \left({\omega }_1\left(t\right)\right)-(\mathsf{\kappa }-2)\varphi \left({\omega }_2\left(t\right)\right)}}.$$

It follows from Parseval’s identity and the maximum modulus principle that

$$\begin{array}{cc}\hfill 2\pi \sum _{n=1}^{\infty }{\left|a_n\right|}^2{r}^{2n}& ={\int }_0^{2\pi }{\left|f\left(t\right)\right|}^{2}d\theta \hfill \\ \hfill & =16{\int }_0^{2\pi }{\left|{\displaystyle \frac{t{D}_{q}f\left(t\right)}{(\mathsf{\kappa }+2)\varphi \left({\omega }_1\left(t\right)\right)-(\mathsf{\kappa }-2)\varphi \left({\omega }_2\left(t\right)\right)}}\right|}^{2}d\theta \hfill \\ \hfill & \ge 16{\int }_0^{2\pi }{\displaystyle \frac{|t{D}_q{f\left(t\right)|}^2}{{\left((\mathsf{\kappa }+2)|\varphi \left({\omega }_1\left(t\right)\right)|+(\mathsf{\kappa }-2)|\varphi \left({\omega }_2\left(t\right)\right)|\right)}^2}}d\theta \hfill \\ \hfill & \ge 16{\int }_0^{2\pi }{\displaystyle \frac{|t{D}_q{f\left(t\right)|}^2}{{\left((\mathsf{\kappa }+2)\underset{|{\omega }_1\left(t\right)|=r}{max}|\varphi \left({\omega }_1\left(t\right)\right)|+(\mathsf{\kappa }-2)\underset{|{\omega }_2\left(t\right)|=r}{max}\left|\varphi \left({\omega }_2\left(t\right)\right)\right|\right)}^2}}d\theta \hfill \\ \hfill & \ge 16{\int }_0^{2\pi }{\displaystyle \frac{|t{D}_q{f\left(t\right)|}^2}{{\left((\mathsf{\kappa }+2)\underset{\left|t\right|=r}{max}\left|\varphi \right({\omega }_1\left(t\right)|+(\mathsf{\kappa }-2)\underset{\left|t\right|=r}{max}\left|\varphi \left({\omega }_2\left(t\right)\right)\right|\right)}^2}}d\theta \hfill \\ \hfill & =16{\int }_0^{2\pi }{\displaystyle \frac{|t{D}_q{f\left(t\right)|}^2}{{\left(2\mathsf{\kappa }\underset{\left|t\right|=r}{max}\left|\varphi \left({\omega }_1\left(t\right)\right)\right|\right)}^2}}d\theta \hfill \\ \hfill & ={\displaystyle \frac{4}{{\left(\mathsf{\kappa }\underset{\left|t\right|=r}{max}\left|\varphi \left({\omega }_1\left(t\right)\right)\right|\right)}^2}}{\int }_0^{2\pi }{\left|t{D}_{q}f\left(t\right)\right|}^{2}d\theta \hfill \\ \hfill & ={\displaystyle \frac{4}{{\left(\mathsf{\kappa }\underset{\left|t\right|=r}{max}\left|\varphi \left({\omega }_1\left(t\right)\right)\right|\right)}^2}}·2\pi \sum _{n=1}^{\infty }{\left[n\right]}_q^2{\left|a_n\right|}^2{r}^{2n}.\hfill \end{array}$$

That is

$$\sum _{n=1}^{\infty }\left(4{\left[n\right]}_q^2-{\mathsf{\kappa }}^2{\left(\underset{\left|t\right|=r}{max}\left|\varphi \left(t\right)\right|\right)}^2\right){\left|a_n\right|}^2\le 0,$$

which gives the desired result. □

The following results are the consequences of Theorem 3.

Corollary 9.

If $f\in {\mathcal{S}}_{\mathsf{\kappa }}^{\ast }\left(\varphi \right)$, then

$$\sum _{n=2}^{\infty }\left(4n^2-{\mathsf{\kappa }}^2{\left(\underset{\left|t\right|=1}{max}\left|\varphi \left(t\right)\right|\right)}^2\right){\left|a_n\right|}^2\le {\mathsf{\kappa }}^2{\left(\underset{\left|t\right|=1}{max}\left|\varphi \left(t\right)\right|\right)}^2-4.$$

Corollary 10.

If $f\in {\mathcal{S}}_q^{\ast }\left(\varphi \right)$, then

$$\sum _{n=2}^{\infty }\left({\left[n\right]}_q^2-{\left(\underset{\left|t\right|=1}{max}\left|\varphi \left(t\right)\right|\right)}^2\right){\left|a_n\right|}^2\le {\left(\underset{\left|t\right|=1}{max}\left|\varphi \left(t\right)\right|\right)}^2-1.$$

When $q\to 1^{-}$ and choose a particular Ma-Minda function $\varphi$, we have the following:

Corollary 11

([17]). If ${\mathcal{S}}_{C_{\mathsf{\kappa }}}^{\ast }$, then

$$\sum _{n=2}^{\infty }\left(4n^2-{\mathsf{\kappa }}^2\left(3+2\sqrt{2}\right)\right){\left|a_n\right|}^2\le {\mathsf{\kappa }}^2\left(3+2\sqrt{2}\right)-4.$$

Corollary 12

([18]). If ${\mathcal{S}}_{L_{\mathsf{\kappa }}}^{\ast }\left(s\right)$, then

$$\sum _{n=2}^{\infty }\left(4n^2-{\mathsf{\kappa }}^2{\left(1+s\right)}^4\right){\left|a_n\right|}^2\le {\mathsf{\kappa }}^2{\left(1+s\right)}^4-4.$$

For $\mathsf{\kappa }=2$ and $\varphi \left(t\right)={\varphi }_6\left(t\right)$, we have

Corollary 13

([9]). If $f\in {\mathcal{S}}^{\ast }\left({\varphi }_6\right):={\mathcal{S}}_{L_{\mathsf{\kappa },q}}^{\ast }\left(s\right)$, then

$$\sum _{n=2}^{\infty }\left(n^2-16{\left({\displaystyle \frac{1+s}{2+(1-q)s}}\right)}^4\right){\left|a_n\right|}^2\le 16{\left({\displaystyle \frac{1+s}{2+(1-q)s}}\right)}^4-1.$$

When $q\to 1^{-}$ and $\mathsf{\kappa }=2$, we have the following results:

Corollary 14

([3]). If $f\in {\mathcal{S}}^{\ast }\left({\varphi }_1\right):={\mathcal{S}}_{\mathrm{e}}^{\ast }$, then

$$\sum _{n=2}^{\infty }\left(n^2-{\mathrm{e}}^2\right){\left|a_n\right|}^2\le {\mathrm{e}}^2-1.$$

Corollary 15

([31]). If $f\in {\mathcal{S}}^{\ast }\left(\varphi \right)$ with $\varphi \left(t\right)=1+4t/3+2t^2/3$, then

$$\sum _{n=2}^{\infty }\left(n^2-9\right){\left|a_n\right|}^2\le 8.$$

3. Conclusions

This article was motivated by the current novel family of the Ma and Minda type, which is related to the functions with bounded radius rotations introduced and studied by Jabeen et al. [24]. In light of this, we began by proving the q-version of the classic Schwarz Pick’s lemma. This result improved significantly from those obtained earlier by Vijaya et al. [29] without proof and Adegani et al. [28]. In addition, we applied this q-classic result to obtain the radius of majorization on the family ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$. Also, under the majorization condition on ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$, bound of the coefficient of $f\in \mathcal{A}$ was obtained. We concluded the findings of this article by obtaining the coefficient bound of $f\in \mathcal{A}$ when the majorization restriction on ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$ was relaxed. Many improvements to the known and some new results were presented as corollaries.

In future work, geometric properties such as integral preservation properties, radius of inclusion properties, and first-order differential subordination relations of ${\mathcal{S}}_{\mathsf{\kappa },q}^{\ast }\left(\varphi \right)$ can be further explored in different directions and perspectives.

It is worth noting that [24] and the present work are the only articles that have first addressed the majorization problem on any family of Ma and Minda type having bounded radius rotations. As a result, these works will serve as the backbone for further investigation in this direction.