On the Fekete–Szegö Problem for Certain Classes of (γ,δ)-Starlike and (γ,δ)-Convex Functions Related to Quasi-Subordinations

Abstract

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates $|a_3-\mu a_2^2|$ for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined.

1. Introduction and Motivation

The conventional analysis is improved by the $\delta$-analysis, which does not require the use of limit notation. The origins of the $\delta$-calculus are reported by Jackson [1,2]. Moreover, numerous applications exist in diverse fields of mathematics and physics (further details can be found in [3,4]). Researchers have recently determined that $(\gamma ,\delta )$-calculus is an extension of $\delta$-calculus. This is because $\delta /\gamma$ is substituted for $\delta$ in $\delta$-calculus. Chakrabarti and Jagannathan [5] (also see [6,7,8]) considered the $(\gamma ,\delta )$-integer. Two essential geometric properties of analytic functions are starlikeness and convexity. The $\delta$-differential operator has thus been used in many articles in Geometric Function Theory. A generalization of starlike functions $S^{*}$ was studied by Ismail et al. [9]. Furthermore, ref. investigated a particular family of $\delta$-Mittag–Leffler functions for closeness to convexity [10]; see also [11,12]. The coefficient inequality of $\delta$-starlike functions was also discussed by [13]. Recently, ref. studied coefficient estimates for $\delta$-convex and $\delta$-starlike functions [14]. As shown and studied in [13,14,15,16,17,18,19], new groups of analytical functions connected with $\delta$-differential operators have also been created and studied. In this work, we extend the concepts of $\delta$-starlikeness and $\delta$-convexity to $(\gamma ,\delta )$-starlikeness and $(\gamma ,\delta )$-convexity. This is based on a recently introduced idea of generalizing $\delta$-analysis to $(\gamma ,\delta )$-analysis. We then obtain the Fekete–Szegö inequality for these classes from this.

These findings will also be used to explain the recently introduced symmetrical $(\gamma ,\delta )$-derivative operator.

Consider $\mathcal{H}$ to be the class of analytic functions defined on the open unit disk $\mathbb{D}=\{\omega \in \mathbb{C}:|\omega |<1\}$ normalized by relations $\zeta \left(0\right)=0$ and ${\zeta }^{\prime }\left(0\right)=1$, with

$$\zeta \left(\omega \right)=\omega +\sum _{m=2}^{\infty }{a}_m{\omega }^m.$$

(1)

Let $\zeta$ and $\mathcal{G}$ be two analytic functions. We say that the function $\zeta$ is subordinate to $\mathcal{G}$ if there exists an analytic function denoted by n such that $n\left(0\right)=0$ and $|n(\omega )|<1$ with property $\zeta \left(\omega \right)=\mathcal{G}\left(n\right(\omega \left)\right)$. We denote this by

$$\zeta \left(\omega \right)\prec \mathcal{G}\left(\omega \right).$$

If the function is univalent in $\mathbb{D}$, then the relation $\zeta \left(\omega \right)\prec \mathcal{G}\left(\omega \right)$ is equivalent to $\zeta \left(0\right)=\mathcal{G}\left(0\right)$ and also $\zeta \left(\mathbb{D}\right)\subset \mathcal{G}\left(\mathbb{D}\right)$.

Note down some definitions and notations from $(\gamma ,\delta )$-calculus theory found in [20]. This is a very interesting paper that motivated our work to improve the results in terms of quasi-subordination and also regarding the generalization of the new classes of analytic functions.

First, the function $\zeta$ has a $(\gamma ,\delta )$-derivative, which is given by

$$D_{\gamma ,\delta }\zeta \left(\omega \right)={\displaystyle \frac{\zeta \left(\gamma \omega \right)-\zeta \left(\delta \omega \right)}{(\gamma -\delta )\omega }}(\omega \ne 0;0<\delta <\gamma \le 1).$$

(2)

It is evident from (2) that if the two functions are $\zeta$ and $\mathcal{G}$, then

$$D_{\gamma ,\delta }(\zeta \left(\omega \right)+\mathcal{G}\left(\omega \right))=D_{\gamma ,\delta }\zeta \left(\omega \right)+D_{\gamma ,\delta }\mathcal{G}\left(\omega \right)$$

(3)

and

$$D_{\gamma ,\delta }\left(cf\left(\omega \right)\right)=c{D}_{\gamma ,\delta }\zeta \left(\omega \right),$$

(4)

where c is a constant.

We note that $D_{\gamma ,\delta }\zeta \left(\omega \right)\to {\zeta }^{\prime }\left(\omega \right)$ as $\gamma =1$ and $\delta \to 1-$, where ${\zeta }^{\prime }$ is the ordinary derivative of the function $\zeta$.

In particular, using (2), the $(\gamma ,\delta )$-derivative of the function $h\left(\omega \right)={\omega }^m$ is as follows:

$$D_{\gamma ,\delta }h\left(\omega \right)={\left[m\right]}_{\gamma ,\delta }{\omega }^{m-1},$$

(5)

where ${\left[m\right]}_{\gamma ,\delta }$ denotes the $(\gamma ,\delta )$-number and is given as:

$${\left[m\right]}_{\gamma ,\delta }={\displaystyle \frac{{\gamma }^m-{\delta }^m}{\gamma -\delta }}(0<\delta <\gamma \le 1).$$

(6)

Considering ${\left[m\right]}_{\gamma ,\delta }\to m$ as $\gamma =1$ and $\delta \to 1-$, we can conclude that, in light of (5), $D_{\gamma ,\delta }h\left(\omega \right)\to h^{\prime }\left(\omega \right)$ as $\gamma =1$ and $\delta \to 1-,$ where $h^{\prime }\left(\omega \right)$ denotes the ordinary derivative of the function $h\left(\omega \right)$ with respect to $\omega$.

Also, using (3), (4) and (5), we deduce the series form of the $(\gamma ,\delta )$-derivative of the function $\zeta$, given by (1) as:

$$D_{\gamma .\delta }\zeta \left(\omega \right)=1+{\Sigma }_{m=2}^{\infty }{\left[m\right]}_{\gamma ,\delta }{a}_m{\omega }^{m-1}(0<\delta <\gamma \le 1)$$

(7)

where ${\left[m\right]}_{\gamma ,\delta }$ denotes the $(\gamma ,\delta )$-number in (6).

Ma and Minda in [21] used the subordination principle between some analytic functions to define the subclasses of starlike functions ${\mathcal{S}}^{*}\left(\psi \right)$ and convex functions $\mathcal{C}\left(\psi \right)$. These subclasses are defined as follows:

$$\begin{array}{cc}\hfill & {\mathcal{S}}^{*}\left(\psi \right)=\left\{\zeta \in \mathcal{H}:{\displaystyle \frac{\omega {\zeta }^{\prime }\left(\omega \right)}{\zeta \left(\omega \right)}}\prec \psi \left(\omega \right)\right\}\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & \mathcal{C}\left(\psi \right)=\left\{\zeta \in \mathcal{H}:\left(1+{\displaystyle \frac{\omega {\zeta }^{″}\left(\omega \right)}{{\zeta }^{\prime }\left(\omega \right)}}\right)\prec \psi \left(\omega \right)\right\},\hfill \end{array}$$

(8)

where $\psi \left(\mathbb{D}\right)$ is symmetric with respect to the real axis and starlike with respect to $\psi \left(0\right)=1$, we have the condition ${\psi }^{\prime }\left(0\right)>0$, and, also, the function $\psi$ is an analytic with a positive real part in $\mathbb{D}$. The function $\zeta \in {\mathcal{S}}^{*}\left(\psi \right)$ is named Ma–Minda starlike (with respect to $\psi$). The paper [22] also defines ${\mathcal{S}}_{\delta }^{*}\left(\psi \right)$ and ${\mathcal{C}}_{\delta }\left(\psi \right)$, respectively, as the classes of $\delta$-starlike and $\delta$-convex functions. They are defined by using the subordination principle as follows:

$$\begin{array}{cc}\hfill & {\mathcal{S}}_{\delta }^{*}\left(\psi \right)=\left\{\zeta \in \mathcal{H}:{\displaystyle \frac{\omega D_{\delta }\left(\omega \right)}{\zeta \left(\omega \right)}}\prec \psi \left(\omega \right)\right\}\hfill \\ \hfill & {\mathcal{C}}_{\delta }\left(\psi \right)=\left\{\zeta \in \mathcal{H}:\left({\displaystyle \frac{D_{\delta }\left(\omega D_{\delta }\zeta \left(\omega \right)\right)}{D_{\delta }\zeta \left(\omega \right)}}\right)\prec \psi \left(\omega \right)\right\},\hfill \end{array}$$

(9)

where the function $\psi \left(\omega \right)$ is analytic in $\mathbb{D}$ such that $\Re \left(\psi \right(\omega \left)\right)>0,\psi \left(0\right)=1$ and ${\psi }^{\prime }\left(0\right)>0$.

Robertson introduces the concept of quasi-subordination in [23]. The function $\zeta$ is quasi-subordinate to $\mathcal{G}$, and we denote it as

$$\zeta \left(\omega \right){\prec }_q\mathcal{G}\left(\omega \right),$$

(10)

if there exist the analytic functions $\phi$ and n such that $|\phi (\omega )|\le 1$, $n\left(0\right)=0$, $|n(\omega )|<1$ and $\zeta \left(\omega \right)=\phi \left(\omega \right)\mathcal{G}\left(n\right(\omega \left)\right)$. Notice that when $\phi \left(\omega \right)=1$, then $\zeta \left(\omega \right)=\mathcal{G}\left(n\right(\omega \left)\right)$; therefore, $\zeta \left(\omega \right)\prec \mathcal{G}\left(\omega \right)$ in $\mathbb{D}$. If condition $n\left(\omega \right)=\omega$ holds, then $\zeta \left(\omega \right)=\phi \left(\omega \right)\mathcal{G}\left(\omega \right)$, and we can say that $\zeta$ is majorized by $\mathcal{G}$ and denote $\zeta \left(\omega \right)\ll \mathcal{G}\left(\omega \right)$ in $\mathbb{D}$. For more information on quasi-subordination, see the following papers: [24,25,26].

Throughout the paper, we assume that $\psi$ is analytic in $\mathbb{D}$, with $\psi \left(0\right)=1$. Motivated by the works [21,22,23,27], we define the following subclasses.

The classes of $(\gamma ,\delta )$-starlike functions and $(\gamma ,\delta )$-convex functions are first defined using the subordination principle. These classes are denoted by ${\mathcal{S}}_{\gamma ,\delta }^{*}\left(\psi \right)$ and ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$, respectively. We do this by substituting the $(\gamma ,\delta )$-derivative for the $\delta$-derivative in the definitions of the corresponding classes of $\delta$-starlike and $\delta$-convex functions.

The following is the definition of the classes ${\mathcal{S}}_{\gamma ,\delta }^{*}\left(\psi \right)$ and ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$:

Definition 1.

Consider the subclass ${\mathcal{S}}_{\gamma ,\delta }^{*}\left(\psi \right)$ containing all functions $\zeta \in \mathcal{H}$ that satisfy the following quasi-subordination:

$${\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}-1{\prec }_q\psi \left(\omega \right)-1.$$

(11)

Example 1.

The function $\zeta :\mathbb{D}\to \mathbb{C}$ defined by the following expression:

$${\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}-1=\omega (\psi \left(\omega \right)-1){\prec }_q\psi \left(\omega \right)-1,$$

belongs to the class ${\mathcal{S}}_{\gamma ,\delta }^{*}\left(\psi \right)$ since

$$\zeta \left(\omega \right)=\omega exp\left(-\omega +{\int }_0^{\omega }\psi \left(\eta \right)d\eta \right).$$

Definition 2.

Consider the class ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$ containing all functions $\zeta \in \mathcal{H}$ satisfying the quasi-subordination

$${\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}{\prec }_q\psi \left(\omega \right)-1.$$

(12)

Example 2.

The function $\zeta :\mathbb{D}\to \mathbb{C}$ given by:

$$\omega D_{\gamma ,\delta }\zeta \left(\omega \right)={\int }_0^{\omega }exp\left(-\eta +{\int }_0^{\eta }\psi \left(\eta \right)d\eta \right)d\eta ,$$

belongs to the class ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$.

The subclasses ${\mathcal{S}}_{\gamma ,\delta }^{*}\left(\psi \right)$ and ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$ are similar to the Ma–Minda starlike and convex classes defined in terms of quasi-subordination.

Definition 3.

Consider the class ${\mathcal{R}}_{\gamma ,\delta }\left(\psi \right)$ consisting of functions $\zeta \in \mathcal{H}$ that satisfy the quasi-subordination relation

$$D_{\gamma ,\delta }\zeta \left(\omega \right)-1{\prec }_q\psi \left(\omega \right)-1.$$

(13)

A function $\zeta \left(\omega \right)\in \mathcal{H}$ with $Re{\zeta }^{\prime }\left(\omega \right)>0$ in D is known to be univalent. We identify this subclass of functions with the class mentioned above of the functions’ positive real part, defined in terms of quasi-subordination.

The functions belonging to the subclasses ${\mathcal{M}}_{\gamma ,\delta }(\sigma ,\psi )$ and ${\mathcal{L}}_{\gamma ,\delta }(\sigma ,\psi )$ are similar to the $\sigma$-convex functions defined by Miller et al. [28] and also to the $\sigma$-logarithmically convex functions defined by Lewandowski et al. [29] (see also [30]).

Definition 4.

Consider the subclass ${\mathcal{M}}_{\gamma ,\delta }(\sigma ,\psi ),(\sigma \ge 0)$ consisting of functions $\zeta \left(\omega \right)\in \mathcal{H}$ satisfying the quasi-subordination

$$(1-\sigma ){\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}+\sigma \left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)-1{\prec }_q\psi \left(\omega \right)-1.$$

(14)

Definition 5.

Consider the class ${\mathcal{L}}_{\gamma ,\delta }(\sigma ,\psi ),(\sigma \ge 0)$ containing all functions $\zeta \left(\omega \right)\in \mathcal{H}$ satisfying the quasi-subordination

$${\left({\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}\right)}^{\sigma }{\left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)}^{1-\sigma }-1{\prec }_q\psi \left(\omega \right)-1.$$

(15)

The m-th coefficient of a univalent function is bounded by m, as stated in [31]. The bounds of coefficients provide information on the different geometric properties of the function. The Fekete–Szegö coefficient bounds for different analytic subclasses have been established by other authors [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. We derive coefficient estimates for the functions in the newly defined classes in the current paper.

Let $\Omega$ be the class of analytic functions n, normalized by $n\left(0\right)=0$ and satisfying the condition $|n(\omega )|<1$. To support our findings, we require the following lemma:

Lemma 1

([47]). If $n\in \Omega$, then we get for any complex number

$$|n_2-t{n}_1^2|\le max\{1;|t|\}.$$

(16)

$$n\left(\omega \right)=n_1\omega +n_2{\omega }^2+\dots ,\omega \in \mathbb{C}$$

The result is sharp for the functions $n\left(\omega \right)={\omega }^2$ or $n\left(\omega \right)=\omega$.

2. Main Results

Because of the importance of the classes ${\mathcal{M}}_{\gamma ,\delta }(\sigma ,\psi )$ and ${\mathcal{L}}_{\gamma ,\delta }(\sigma ,\psi )$, Theorems 1 and 4 are stated and proved separately.

Consider $\zeta \left(\omega \right)=\omega +a_2{\omega }^2+a_3{\omega }^3+\dots ,$ $\psi \left(\omega \right)=1+{\mathcal{B}}_1\omega +{\mathcal{B}}_2{\omega }^2+{\mathcal{B}}_3{\omega }^3+\dots ,$ $\phi \left(\omega \right)=c_0+c_1\omega +c_2{\omega }^2+c_3{\omega }^3+\dots ,$ ${\mathcal{B}}_1\in \mathbb{R}\mathrm{and}{\mathcal{B}}_1>0.$

Theorem 1.

If $\zeta \in \mathcal{H}$ belongs to ${\mathcal{S}}_{\gamma ,\delta }^{*}\left(\psi \right)$, then

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }-1}},\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left({\mathcal{B}}_1+max\left\{{\mathcal{B}}_1,{\displaystyle \frac{{\mathcal{B}}_1^2}{{\left[2\right]}_{\gamma ,\delta }-1}}+|{\mathcal{B}}_2|\right\}\right),\hfill \end{array}$$

(17)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left(1+max\left\{1,\left|\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right)\right|{\mathcal{B}}_1+\left|{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\right)\right\}.$$

(18)

Proof. 

Since $\zeta \left(\omega \right)\in S_{\gamma ,\delta }^{*}\left(\psi \right),$ then there exist two analytic functions $\phi$ and n, with $|\phi (\omega )|\le 1,n\left(0\right)=0$ and $|n(\omega )|<1$ such that

$${\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}-1=\phi \left(\omega \right)(\psi \left(n\left(\omega \right)\right)-1).$$

(19)

Since

$$\begin{array}{cc}\hfill {\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}-1& =({\left[2\right]}_{\gamma ,\delta }-1)a_2\omega +\left(({\left[3\right]}_{\gamma ,\delta }-1)a_3-({\left[2\right]}_{\gamma ,\delta }-1)a_2^2\right){\omega }^2+\dots ,\hfill \\ \hfill \psi \left(n\right(\omega \left)\right)-1& ={\mathcal{B}}_1{n}_1\omega +\left({\mathcal{B}}_1{n}_2+{\mathcal{B}}_2{n}_1^2\right){\omega }^2+\dots .\hfill \end{array}$$

(20)

$$\phi \left(\omega \right)(\psi \left(n\left(\omega \right)\right)-1)={\mathcal{B}}_1{c}_0{n}_1\omega +\left({\mathcal{B}}_1{c}_1{n}_1+c_0\left({\mathcal{B}}_1{n}_2+{\mathcal{B}}_2{n}_1^2\right)\right){\omega }^2+\dots ,$$

(21)

it follows from (19) that

$$\begin{array}{cc}\hfill & a_2={\displaystyle \frac{{\mathcal{B}}_1{c}_0{n}_1}{{\left[2\right]}_{\gamma ,\delta }-1}}\hfill \\ \hfill a_3& ={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left({\mathcal{B}}_1{c}_1{n}_1+{\mathcal{B}}_1{c}_0{n}_2+c_0\left({\mathcal{B}}_2+{\displaystyle \frac{{\mathcal{B}}_1^2{c}_0}{{\left[2\right]}_{\gamma ,\delta }-1}}\right)n_1^2\right).\hfill \end{array}$$

(22)

Since $\phi$ is analytic and bounded in $\mathbb{D}$, we get [49] (p. 172)

$$|c_m|\le 1-|c_0{|}^2\le 1(m>0).$$

(23)

By using this fact and the well-known inequality $|n_1|\le 1$, we get

$$|a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }-1}}.$$

(24)

Further,

$$a_3-\alpha a_2^2={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left({\mathcal{B}}_1{c}_1{n}_1+{\mathcal{B}}_1{c}_0{n}_2+c_0\left({\mathcal{B}}_2+{\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}{\mathcal{B}}_1^2{c}_0-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha {\mathcal{B}}_1^2{c}_0\right)n_1^2\right).$$

(25)

Then

$$\left|a_3-\alpha a_2^2\right|\le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left(|{\mathcal{B}}_1{c}_1{n}_1|+\left|{\mathcal{B}}_1{c}_0\left(n_2-\left({\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha {\mathcal{B}}_1{c}_0-{\displaystyle \frac{{\mathcal{B}}_1{c}_0}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right)n_1^2\right)\right|\right).$$

(26)

Again applying $|c_m|\le 1$ and $|n_1|\le 1$, we have

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left(1+\left|\left(n_2-\left(-\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right){\mathcal{B}}_1{c}_0-{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right)n_1^2\right)\right|\right).$$

(27)

Applying Lemma 1 to

$$\begin{array}{cc}\hfill & \left|\left(n_2-\left(-\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right){\mathcal{B}}_1{c}_0-{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right)n_1^2\right)\right|\hfill \\ & \le max\{1;\left|-\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right){\mathcal{B}}_1{c}_0-{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right\}\hfill \end{array}$$

(28)

yields

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left(1+max\left\{1,\left|-\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right){\mathcal{B}}_1{c}_0-{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\right)\right\}.$$

(29)

Observe that

$$\left|-\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right){\mathcal{B}}_1{c}_0-{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\le {\mathcal{B}}_1|c_0|\left|\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right)\right|+\left|{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|,$$

(30)

and hence, we can conclude that

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left(1+max\left\{1,\left|\left({\displaystyle \frac{1}{{\left[2\right]}_{\gamma ,\delta }-1}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }-1}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right)\right|{\mathcal{B}}_1+\left|{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\right)\right\}.$$

(31)

For $\alpha =0$, the above will reduce the estimate of $|a_3|$. □

Remark 1.

1. 

For $\phi \left(\omega \right)\equiv 1$, Theorem 1 offers a particular case of the evaluation in Theorem 1 in [34] for $\gamma =1,\delta =1$ and Theorem 1 in [35] for $k=1$.

2. 

When $\gamma =1,\delta \to -1,$ $\phi \left(\omega \right)\equiv 1$, we get the results of [20].

Theorem 2.

Let $\zeta \in \mathcal{H}$ and satisfying

$${\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}-1\ll \psi \left(\omega \right)-1,$$

(32)

Consequently, the subsequent inequalities are valid:

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }-1}},\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left({\mathcal{B}}_1+{\displaystyle \frac{{\mathcal{B}}_1^2}{{\left[2\right]}_{\gamma ,\delta }-1}}+|{\mathcal{B}}_2|\right),\hfill \end{array}$$

(33)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|\le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }-1}}\left({\mathcal{B}}_1+\left|{\displaystyle \frac{1}{({\left[2\right]}_{\gamma ,\delta }-1)}}-{\displaystyle \frac{({\left[3\right]}_{\gamma ,\delta }-1)}{{({\left[2\right]}_{\gamma ,\delta }-1)}^2}}\alpha \right|{\mathcal{B}}_1^2+\left|{\mathcal{B}}_2\right|\right).$$

(34)

Proof. 

We proceed by taking $n\left(\omega \right)=\omega$ in the proof of Theorem 1. □

Theorem 3.

If $\zeta \in \mathcal{H}$ belongs to ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$, then

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)}}\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)}}\left({\mathcal{B}}_1+max\left\{{\mathcal{B}}_1,{\displaystyle \frac{{\mathcal{B}}_1^2}{({\left[2\right]}_{\gamma ,\delta }-1)}}+|{\mathcal{B}}_2|\right\}\right),\hfill \end{array}$$

(35)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)}}\left(1+max\left\{1,\left|\left({\displaystyle \frac{1}{({\left[2\right]}_{\gamma ,\delta }-1)}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)}{{\left({\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)\right)}^2}}\alpha \right)\right|{\mathcal{B}}_1+\left|{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\right)\right\}.$$

(36)

Proof. 

Observe that when $\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\in {\mathcal{S}}_{\gamma ,\delta }^{*}$, equality (19) becomes

$${\displaystyle \frac{\omega D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}}-1=\phi \left(\omega \right)(\psi \left(n\left(\omega \right)\right)-1),$$

(37)

or, equally,

$${\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\prec \psi \left(n\left(\omega \right)\right)-1,$$

(38)

and the converse can be verified easily. By the Alexander relation—that is, ${\mathcal{C}}_{\gamma ,\delta }\left(\psi \right)$ if and only if $\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\in {\mathcal{S}}_{\gamma ,\delta }^{*}$—we can obtain the required estimates. □

Theorem 4.

If $\zeta \left(\omega \right)\in \mathcal{H}$ satisfies

$${\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\ll \psi \left(\omega \right)-1,$$

(39)

consequently, the subsequent inequalities are valid:

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)}}\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)}}\left({\mathcal{B}}_1+{\displaystyle \frac{{\mathcal{B}}_1^2}{({\left[2\right]}_{\gamma ,\delta }-1)}}+|{\mathcal{B}}_2|\right),\hfill \end{array}$$

(40)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)}}\left({\mathcal{B}}_1+\left|{\displaystyle \frac{1}{({\left[2\right]}_{\gamma ,\delta }-1)}}-{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)}{{\left({\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)\right)}^2}}\alpha \right|{\mathcal{B}}_1^2+\left|{\mathcal{B}}_2\right|\right).$$

(41)

Theorem 5.

If $\zeta \left(\omega \right)\in \mathcal{H}$ belongs to ${\mathcal{R}}_{\gamma ,\delta }\left(\psi \right)$, then

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }}}\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+max\left\{{\mathcal{B}}_1,|{\mathcal{B}}_2|\right\}\right),\hfill \end{array}$$

(42)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+max\left\{{\mathcal{B}}_1,{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }}{{\left[2\right]}_{\gamma ,\delta }^2}}{\mathcal{B}}_1^2|\alpha |+|{\mathcal{B}}_2|\right\}\right).$$

(43)

Proof. 

For $\zeta \left(\omega \right)\in {\mathcal{R}}_{\gamma ,\delta }\left(\psi \right)$, we know that by Definition 3 there exist analytic functions $\phi$ and n, with $|\phi (\omega )|\le 1$, $n\left(0\right)=0$ and $|n(\omega )|<1$, such that

$$D_{\gamma ,\delta }\zeta \left(\omega \right)-1=\phi \left(\omega \right)(\psi \left(n\left(\omega \right)\right)-1).$$

(44)

Since

$$D_{\gamma ,\delta }\zeta \left(\omega \right)-1={\left[2\right]}_{\gamma ,\delta }{a}_2\omega +{\left[3\right]}_{\gamma ,\delta }{a}_3{\omega }^2+\dots ,$$

(45)

it follows from (44) and (21) that

$$\begin{array}{cc}\hfill & a_2={\displaystyle \frac{{\mathcal{B}}_1{c}_0{n}_1}{{\left[2\right]}_{\gamma ,\delta }}}\hfill \\ \hfill a_3& ={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1{c}_1{n}_1+c_0\left({\mathcal{B}}_1{n}_2+{\mathcal{B}}_2{n}_1^2\right)\right).\hfill \end{array}$$

(46)

We can write that

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }}}\hfill \\ \hfill \left|a_3-\alpha a_2^2\right|& \le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }}}(1+\left|n_2-\left({\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }{\mathcal{B}}_1{c}_0}{{\left[2\right]}_{\gamma ,\delta }^2}}\alpha -{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right)n_1^2\right|\hfill \end{array}$$

(47)

following the same argument as is presented in Theorem 1, where $|c_0|\le 1$ and $|c_1|\le 1$.

Applying Lemma 1, we get

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{{\mathcal{B}}_1}{{\left[3\right]}_{\gamma ,\delta }}}\left(1+max\left\{1,\left|{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }{\mathcal{B}}_1{c}_0}{{\left[2\right]}_{\gamma ,\delta }^2}}\alpha -{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\right\}\right).$$

(48)

Since

$$\left|{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }{\mathcal{B}}_1{c}_0}{{\left[2\right]}_{\gamma ,\delta }^2}}\alpha -{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|\le {\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }^2}}|c_0||\alpha |+\left|{\displaystyle \frac{{\mathcal{B}}_2}{{\mathcal{B}}_1}}\right|$$

(49)

and $|c_0|\le 1$, we can conclude the hypothesis. □

Theorem 6.

If $\zeta \left(\omega \right)\in \mathcal{H}$ satisfies

$$D_{\gamma ,\delta }\zeta \left(\omega \right)-1\ll \psi \left(\omega \right)-1,$$

(50)

consequently, the subsequent inequalities are valid:

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }}},\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+|{\mathcal{B}}_2|\right),\hfill \end{array}$$

(51)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }}{{\left[2\right]}_{\gamma ,\delta }^2}}{\mathcal{B}}_1^2|\alpha |+\left|{\mathcal{B}}_2\right|\right).$$

(52)

Let the class ${\mathcal{R}}_{\gamma ,\delta }^{\rho }\left(\psi \right)$ consist of functions $\zeta \in \mathcal{H}$ satisfying the quasi-subordination

$${\displaystyle \frac{1}{\rho }}(D_{\gamma ,\delta }\zeta \left(\omega \right)-1){\prec }_q\psi \left(\omega \right)-1,$$

(53)

where $\rho \in \mathbb{C}\setminus \left\{0\right\}$. The following corollary gives the results for $\zeta \in {\mathcal{R}}_{\gamma ,\delta }^{\rho }\left(\psi \right).$

Corollary 1.

Let $\rho \in \mathbb{C}\setminus \left\{0\right\}$. If $\zeta \left(\omega \right)\in \mathcal{H}$ belongs to ${\mathcal{R}}_{\gamma ,\delta }^{\rho }\left(\psi \right),$ then

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{|\rho |}{{\left[2\right]}_{\gamma ,\delta }}}{\mathcal{B}}_1,\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{|\rho |}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+|{\mathcal{B}}_2|\right),\hfill \end{array}$$

(54)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{|\rho |}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }}{{\left[2\right]}_{\gamma ,\delta }^2}}{\mathcal{B}}_1^2|\alpha |+\left|{\mathcal{B}}_2\right|\right).$$

(55)

Remark 2.

1. 

For $\phi \left(\omega \right)\equiv 1$, Corollary 1 gives a particular case of the estimates in Theorem 3 in [34] for $\gamma =1,\delta =1$ and Theorem 3 in [35] for $k=1$.

2. 

For $\phi \left(\omega \right)\equiv 1$ and $\psi \left(\omega \right)=(1+A\omega )/(1+\mathcal{B}\omega ),(-1\le \mathcal{B}<A\le 1)$, Corollary 1 reduces to the results in Theorem 4 in [40].

3. 

When $\gamma =1,\delta \to -1,$ $\phi \left(\omega \right)\equiv 1$, we get the results of [20].

Theorem 7.

Let $\sigma \ge 0.$ If $\zeta \left(\omega \right)\in \mathcal{H}$ belongs to ${\mathcal{M}}_{\gamma ,\delta }(\sigma ,\psi )$, then

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{{\left[2\right]}_{\gamma ,\delta }}}\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+max\left\{{\mathcal{B}}_1,|{\mathcal{B}}_2|\right\}\right),\hfill \end{array}$$

(56)

and for any complex number α,

$$\left|a_3-\alpha a_2^2\right|={\displaystyle \frac{1}{{\left[3\right]}_{\gamma ,\delta }}}\left({\mathcal{B}}_1+max\left\{{\mathcal{B}}_1,{\displaystyle \frac{{\left[3\right]}_{\gamma ,\delta }}{{\left[2\right]}_{\gamma ,\delta }^2}}{\mathcal{B}}_1^2|\alpha |+|{\mathcal{B}}_2|\right\}\right).$$

(57)

Proof. 

If $\zeta \left(\omega \right)\in {\mathcal{M}}_{\gamma ,\delta }(\sigma ,\psi )$ for $\sigma \ge 0$, then there are analytic functions $\phi$ and n, with $|\phi (\omega )|\le 1,n\left(0\right)=0$ and $|n(\omega )|<1$, such that

$$(1-\sigma ){\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}+\sigma \left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)-1=\phi \left(\omega \right)(\psi \left(n\left(\omega \right)\right)-1).$$

(58)

According to a computation,

$$\begin{array}{cc}\hfill (1-\sigma ){\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}& =(1-\sigma )+(1-\sigma )({\left[2\right]}_{\gamma ,\delta }-1)a_2\omega +(1-\sigma )\left(({\left[3\right]}_{\gamma ,\delta }-1)a_3-({\left[2\right]}_{\gamma ,\delta }-1)a_2^2\right){\omega }^2+\dots ,\hfill \\ \hfill \sigma \left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)& =\sigma +\sigma {\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)a_2\omega +\sigma \left({\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)a_3-{\left[2\right]}_{\gamma ,\delta }^2({\left[2\right]}_{\gamma ,\delta }-1)a_2^2\right){\omega }^2+\dots .\hfill \end{array}$$

(59)

Hence, from (59), we have

$$\begin{array}{cc}\hfill & (1-\sigma ){\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}+\sigma \left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)-1=({\left[2\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[2\right]}_{\gamma ,\delta }-1))a_2\omega +\hfill \\ & \left(({\left[3\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[3\right]}_{\gamma ,\delta }-1))a_3-({\left[2\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[2\right]}_{\gamma ,\delta }^2-1))a_2^2\right){\omega }^2+\dots .\hfill \end{array}$$

(60)

It then follows from relations (58) and (21) that

$$\begin{array}{cc}\hfill a_2& ={\displaystyle \frac{{\mathcal{B}}_1{c}_0{n}_1}{({\left[2\right]}_{\gamma ,\delta }-1))(1+\sigma ({\left[2\right]}_{\gamma ,\delta }-1))}}\hfill \\ \hfill a_3& ={\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[3\right]}_{\gamma ,\delta }-1))}}\left({\mathcal{B}}_1{c}_1{n}_1+{\mathcal{B}}_1{c}_0{n}_2+\left({\mathcal{B}}_2{c}_0+{\displaystyle \frac{{\mathcal{B}}_1^2{c}_0^2}{({\left[2\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[2\right]}_{\gamma ,\delta }^2-1))}}\right)n_1^2\right).\hfill \end{array}$$

(61)

The proof can be concluded by following a procedure similar to that employed in prior theorems. □

Remark 3.

1. 

Considering $\phi \left(\omega \right)\equiv 1$, Corollary 1 gives a particular case of the estimates in Theorem 3 in [34] for $\gamma =1,\delta =1$ and Theorem 3 in [35] for $k=1$.

2. 

By taking $\phi \left(\omega \right)\equiv 1$ and $\psi \left(\omega \right)=(1+A\omega )/(1+\mathcal{B}\omega ),(-1\le \mathcal{B}<A\le 1),$ Corollary 1 reduces to the results in Theorem 4 in [40].

3. 

When $\gamma =1,\delta \to -1,$ $\phi \left(\omega \right)\equiv 1$, we get the results of [20].

Theorem 8.

Let $\sigma \ge 0.$ If $\zeta \left(\omega \right)\in \mathcal{A}$ satisfies

$$(1-\sigma ){\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}+\sigma \left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)-1\ll \psi \left(\omega \right)-1,$$

(62)

Consequently, the subsequent inequalities are valid:

$$\begin{array}{cc}\hfill |a_2|& \le {\displaystyle \frac{{\mathcal{B}}_1}{({\left[2\right]}_{\gamma ,\delta }-1))(1+\sigma ({\left[2\right]}_{\gamma ,\delta }-1))}}\hfill \\ \hfill |a_3|& \le {\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[3\right]}_{\gamma ,\delta }-1))}}\left({\mathcal{B}}_1+|{\mathcal{B}}_2|+{\displaystyle \frac{{\mathcal{B}}_1^2}{({\left[2\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[2\right]}_{\gamma ,\delta }^2-1))}}\right),\hfill \end{array}$$

(63)

and for any complex number α,

$$|a_3-\alpha a_2^2|\le {\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[3\right]}_{\gamma ,\delta }-1))}}\left({\mathcal{B}}_1+|{\mathcal{B}}_2|+{\displaystyle \frac{\left|({\left[3\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[3\right]}_{\gamma ,\delta }-1))\alpha -1\right|}{({\left[2\right]}_{\gamma ,\delta }-1)(1+\sigma ({\left[2\right]}_{\gamma ,\delta }^2-1))}}{\mathcal{B}}_1^2\right).$$

(64)

Theorem 9.

Let $\sigma \ge 0$ and $\nu =1-\sigma .$ If $\zeta \in \mathcal{H}$ belongs to ${\mathcal{L}}_{\gamma ,\delta }(\sigma ,\psi ),$ then

$$\begin{array}{cc}\hfill |a_2|& \le {\displaystyle \frac{{\mathcal{B}}_1}{({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )}}\hfill \\ \hfill & |a_3|\le {\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +\nu {\left[3\right]}_{\gamma ,\delta })}}\left({\mathcal{B}}_1\right.+\hfill \\ & \left.max\left\{{\mathcal{B}}_1,{\displaystyle \frac{|{\left(\sigma ({\left[2\right]}_{\gamma ,\delta }-1)+\nu {\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)\right)}^2-\left(3\sigma ({\left[2\right]}_{\gamma ,\delta }-1)+\nu {\left[2\right]}_{\gamma ,\delta }^2({\left[2\right]}_{\gamma ,\delta }^2-1)\right)|}{2{\left(({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )\right)}^2}}{\mathcal{B}}_1^2+|{\mathcal{B}}_2|\right\}\right).\hfill \end{array}$$

(65)

and for any complex number α,

$$\begin{array}{cc}\hfill & \left|a_3-\alpha a_2^2\right|\le {\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +\nu {\left[3\right]}_{\gamma ,\delta })}}\left({\mathcal{B}}_1\right.+max\left\{{\mathcal{B}}_1,\right.\hfill \\ & \left.\left.{\displaystyle \frac{|{\left(\sigma ({\left[2\right]}_{\gamma ,\delta }-1)+\nu {\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)\right)}^2-\left(3\sigma ({\left[2\right]}_{\gamma ,\delta }-1)+\nu {\left[2\right]}_{\gamma ,\delta }^2({\left[2\right]}_{\gamma ,\delta }^2-1)\right)-2\alpha ({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +\nu {\left[3\right]}_{\gamma ,\delta })|}{2{\left(({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )\right)}^2}}{\mathcal{B}}_1^2+|{\mathcal{B}}_2|\right\}\right).\hfill \end{array}$$

(66)

Proof. 

If $\zeta \left(\omega \right)\in {\mathcal{L}}_{\gamma ,\delta }(\sigma ,\psi )$ for $\sigma \ge 0$ and $\nu =1-\sigma$, then there are analytic functions $\phi$ and n, with $|\phi (\omega )|\le 1$, $n\left(0\right)=0$ and $|n(\omega )|<1$, such that

$${\left({\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}\right)}^{\sigma }{\left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)}^{\nu }-1=\phi \left(\omega \right)(\psi \left(n\left(\omega \right)\right)-1).$$

(67)

According to a computation,

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}\right)}^{\sigma }=1+& \sigma ({\left[2\right]}_{\gamma ,\delta }-1)a_2\omega +{\displaystyle \frac{\sigma }{2}}\left[2({\left[3\right]}_{\gamma ,\delta }-1)a_3-({\left[2\right]}_{\gamma ,\delta }-1)\left(1+{\left[2\right]}_{\gamma ,\delta }-\sigma ({\left[2\right]}_{\gamma ,\delta }-1)\right)a_2^2\right]{\omega }^2+\dots ,\hfill \\ \hfill {\left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)}^{\nu }& =1+\nu \left({\left[2\right]}_{\gamma ,\delta }({\left[2\right]}_{\gamma ,\delta }-1)\right)a_2\omega +\hfill \\ \hfill & {\displaystyle \frac{\nu }{2}}\left[2({\left[3\right]}_{\gamma ,\delta }({\left[3\right]}_{\gamma ,\delta }-1)a_3-{\left[2\right]}_{\gamma ,\delta }^2({\left[2\right]}_{\gamma ,\delta }-1)\left(1+{\left[2\right]}_{\gamma ,\delta }-\nu ({\left[2\right]}_{\gamma ,\delta }-1)\right)a_2^2\right]{\omega }^2+\dots .\hfill \end{array}$$

(68)

Thus, (68) gives

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}\right)}^{\sigma }{\left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)}^{\nu }-1=({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )a_2\omega +\hfill \\ & \left(({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +{\left[3\right]}_{\gamma ,\delta }\nu )a_3-\left({\displaystyle \frac{{\left[2\right]}_{\gamma ,\delta }-1}{2}}\right)\left[(1+{\left[2\right]}_{\gamma ,\delta })(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )-({\left[2\right]}_{\gamma ,\delta }-1){(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )}^2\right]a_2^2\right){\omega }^2+\dots ,\hfill \end{array}$$

(69)

By using the above equation and (21) in (67), we have

$$\begin{array}{cc}\hfill a_2& ={\displaystyle \frac{{\mathcal{B}}_1{c}_0{n}_1}{({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )}}\hfill \\ \hfill a_3& ={\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +{\left[3\right]}_{\gamma ,\delta }\nu )}}\left({\mathcal{B}}_1{c}_1{n}_1+{\mathcal{B}}_1{c}_0{n}_2\right.+\hfill \\ & \left.\left({\mathcal{B}}_2{c}_0-{\displaystyle \frac{\left(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu \left)\right({\left[2\right]}_{\gamma ,\delta }-1\right)-\left({\left[2\right]}_{\gamma ,\delta }+1\right)}{2({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )}}{\mathcal{B}}_1^2{c}_0^2\right)n_1^2\right).\hfill \end{array}$$

(70)

We can prove the hypothesis in the same way as we did with earlier theorems. □

Remark 4.

1. 

Theorem 9 may be simplified to Theorem 3 when $\sigma =0$ and $\nu =1$.

2. 

Theorem 9 may be simplified to Theorem 1 when $\sigma =1$ and $\nu =0$.

3. 

For the equation $\phi \left(\omega \right)\equiv 1$, Theorem 9 is a specific instance of the estimates presented in Theorem 7 in [35] for the value of $k=1$.

4. 

When $\gamma =1,\delta \to -1,$ $\phi \left(\omega \right)\equiv 1$, we get the results of [20].

Theorem 10.

Let $\sigma \ge 0$ and $\nu =1-\sigma .$ If $\zeta \in \mathcal{H}$ satisfies

$${\left({\displaystyle \frac{\omega D_{\gamma ,\delta }\zeta \left(\omega \right)}{\zeta \left(\omega \right)}}\right)}^{\sigma }{\left({\displaystyle \frac{D_{\gamma ,\delta }\left(\omega D_{\gamma ,\delta }\zeta \left(\omega \right)\right)}{D_{\gamma ,\delta }\zeta \left(\omega \right)}}\right)}^{1-\sigma }-1\ll \psi \left(\omega \right)-1,$$

(71)

consequently, the subsequent inequalities are valid:

$$\begin{array}{cc}\hfill & |a_2|\le {\displaystyle \frac{{\mathcal{B}}_1}{({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )}},\hfill \\ \hfill & |a_3|\le {\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +{\left[3\right]}_{\gamma ,\delta }\nu )}}\left({\mathcal{B}}_1+{\displaystyle \frac{\left|\left(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )({\left[2\right]}_{\gamma ,\delta }-1)\right)-\left({\left[2\right]}_{\gamma ,\delta }+1\right)\right|}{2({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )}}{\mathcal{B}}_1^2+|{\mathcal{B}}_2|\right),\hfill \end{array}$$

(72)

and for any complex number α,

$$\begin{array}{cc}\hfill & \left|a_3-\alpha a_2^2\right|\le {\displaystyle \frac{1}{({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +\nu {\left[3\right]}_{\gamma ,\delta })}}\left({\mathcal{B}}_1\right.+\hfill \\ & \left.{\displaystyle \frac{\left|[(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )({\left[2\right]}_{\gamma ,\delta }-1)-({\left[2\right]}_{\gamma ,\delta }+1)]({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )-2\alpha ({\left[3\right]}_{\gamma ,\delta }-1)(\sigma +{\left[3\right]}_{\gamma ,\delta }\nu )\right|}{2{\left(({\left[2\right]}_{\gamma ,\delta }-1)(\sigma +{\left[2\right]}_{\gamma ,\delta }\nu )\right)}^2}}{\mathcal{B}}_1^2+|{\mathcal{B}}_2|\right).\hfill \end{array}$$

(73)

3. Conclusions

In the present paper, with the extended idea of a symmetric $(\gamma ,\delta )$-derivative operator, we study two subclasses of functions. For certain values of the parameters, we obtain some special classes studied earlier by various authors. The new results estimate the upper bound coefficients expressed in the Fekete–Szegö problem. We also deduce results that generalize and improve several previously known ones. We obtain certain applications of the main results for the $(\gamma ,\delta )$-starlike and $(\gamma ,\delta )$-convex functions by applying the $(\gamma ,\delta )$-derivative operator. In addition, we also provide certain applications to support our results. Thus, an interesting aspect of the paper is that the generalized classes of $(\gamma ,\delta )$-starlike and $(\gamma ,\delta )$-convex functions, which are a generalization of starlike and convex functions, are established by employing the $(\gamma ,\delta )$-derivative operator. We still intend to continue work on the present issue. In this direction, for future research:

• We plan to study novel symmetric $(\gamma ,\delta )$-derivative operators;
• We will target the derivation of new aspects of the Fekete–Szegö problem for quasi-subordination;
• We will take into account the establishment of correlations between the findings presented here and similar results in the field.

This research’s findings have potential applications in the fields of post-quantum theory and symmetry. We anticipate that the novel concepts and innovative methodologies presented in this work will captivate readers with a keen interest in the realm of geometric function theory.