Yamaguchi -Noshiro Type Bi-Univalent Functions Associated with Sălăgean-Erdély–Kober Operator

Abstract

We defined two new subclasses of analytic bi-univalent function class $\Sigma ,$ in the open unit disk related with the Sălăgean–Erdély–Kober operator. The bounds on initial coefficients $|a_2|,|a_3|$ and $|a_4|$ for the functions in these new subclasses of $\Sigma$ are investigated. Using the estimates of coefficients $a_2,a_3$, we also discuss the Fekete-Szegö inequality results for the function classes defined in this paper. Relevant connections of these results, presented here as corollaries, are new and not studied in association with Sălăgean-Erdély–Kober operator for the subclasses defined earlier.

1. Introduction

Let $\mathcal{A}$ denote the class of holomorphic functions in open unit disk

$$\Delta :=\{z:z\in \mathbb{C}\mathrm{and}|z|<1\}$$

be given by the power series

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

(1)

satisfying normalization conditions (see [1])

$$f\left(0\right)=f^{\prime }\left(0\right)-1=0,$$

normalization $f\left(0\right)=0$ geometrically amounts to only a translation of the image domain, and $f^{\prime }\left(0\right)=1$ corresponds to the rotation of the image domain.The subclass of $\mathcal{A}$ consisting of all univalent functions f in $\Delta$ is denoted by $\mathcal{S}.$

Let $f_1,f_2\in \mathcal{A}$ and be assumed as

$$f_1\left(z\right)=\sum _{n=0}^{\infty }{a}_{n,1}{z}^n\mathrm{and}g\left(z\right)=\sum _{n=0}^{\infty }{a}_{n,2}{z}^n(z\in \Delta ),$$

are convoluted $f_1\ast f_2$ if the product is defined as

$$(f_1\ast f_2)\left(z\right)=\sum _{n=0}^{\infty }{a}_{n{,}_1}{a}_{n,2}{z}^n=(f_2\ast f_1)\left(z\right)(z\in \Delta ).$$

Recently, several authors have contributed to the growth of fractional calculus (differentiation and integration of arbitrary orders). Fractional calculus often find its applications in the field of engineering, such as capacitor theory, electrode–electrolyte interface models, feedback amplifiers, generalized voltage dividers, fractional order models of neurons, the electric conductance of biological systems, fitting experimental data, medical, and memory characteristics [2,3,4,5,6]. Fractional derivative operators, which are frequently defined through fractional integral operators, help in gathering useful information about the progress of the resources and processes involved in the phenomena. Many fractional derivative operators, such as the Riemann–Liouville fractional derivative operator associated with hypergeometric type function, the Caputo (CF) and Erdélyi–Kober (EK) fractional operators have been proposed and studied extensively in the literature. The Riemann–Liouville (R–L) fractional integral operator of order $\epsilon >0$, which is one of the most used and studied (see [2,3,4,5,6,7]) operators, is given by:

$${\mathcal{V}}_{{ϵ}^{+}}^{\epsilon }f\left(t\right)=\frac{1}{\Gamma (\epsilon )}\underset{ϵ}{\overset{t}{\int }}{(t-\kappa )}^{\epsilon -1}f\left(\kappa \right)d\kappa ,t>ϵ.$$

First, we recall the following differential operators:

In 1983, Sălăgean [8] introduced differential operator ${\mathcal{D}}^m:\mathcal{A}\to \mathcal{A}$ defined by

$$\begin{array}{cc}\hfill {\mathcal{D}}^{0}f\left(z\right)& =f\left(z\right),{\mathcal{D}}^{1}f\left(z\right)=\mathcal{D}f\left(z\right)=z{f}^{\prime }\left(z\right),\hfill \\ \hfill {\mathcal{D}}^{m}f\left(z\right)& =\mathcal{D}{\left({\mathcal{D}}^{m-1}f\left(z\right)\right)}^{\prime }=z{\left({\mathcal{D}}^{m-1}f\left(z\right)\right)}^{\prime },m\in \mathbb{N}=\{1,2,3,\cdots \}.\hfill \end{array}$$

We note that

$${\mathcal{D}}^{m}f\left(z\right)=z+\sum _{n=2}^{\infty }{n}^m{a}_n{z}^n,m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}.$$

(2)

Let the integral operator be

$$I_{\ell }^{\varrho }:\mathcal{A}\to \mathcal{A}$$

by

$$I_{\ell }^{\varrho }f\left(z\right)=\left\{\begin{array}{c}\frac{\ell +1}{\varrho }{z}^{1-\frac{\ell +1}{\varrho }}\underset{0}{\overset{z}{\int }}{t}^{\frac{\ell +1}{\varrho }-2}f\left(t\right)dt;\varrho \ne 0,\\ f\left(z\right),\varrho =0.\end{array}\right.$$

The differential operator is

$$D_{\ell }^{\varrho }:\mathcal{A}\to \mathcal{A}$$

by

$$D_{\ell }^{\varrho }f\left(z\right)=\left\{\begin{array}{c}\frac{\varrho }{\ell +1}{z}^{2-\frac{\ell +1}{\varrho }}\frac{d}{dz}\left(z^{\frac{\ell +1}{\varrho }-1}f\left(z\right)\right);\varrho \ne 0,\\ f\left(z\right),\varrho =0.\end{array}\right.$$

For $\ell >-1;\varrho >0,$ and $m\in \mathbb{Z}=\{0,\pm 1,\pm 2,\pm 3,\cdots \};$ let

$$D_{\ell ,\varrho }^m:\mathcal{A}\to \mathcal{A}$$

by

$$D_{\ell ,\varrho }^{m}f\left(z\right)=\left\{\begin{array}{cc}{D}_{\ell }^{\varrho }{\mathcal{D}}_{\ell ,\varrho }^{m-1}f\left(z\right)& m\in {\mathbb{Z}}^{+};\\ I_{\ell }^{\varrho }{\mathcal{D}}_{\ell ,\varrho }^{m+1}f\left(z\right),& m\in {\mathbb{Z}}^{-};\\ f\left(z\right),& m=0.\end{array}\right.$$

The following Erdély–Kober type ([9] (Section 5)) integral operator is used throughout this paper:

Definition 1.

Erdély–Kober operator (EK): Let for $\vartheta >0,a,c\in \mathbb{C}$, be such that $\Re (c-a)\ge 0,$ and $\mathcal{R}\left(a\right)>-\vartheta$ an Erdély–Kober type integral operator is

$${\mathcal{V}}_{\vartheta }^{a,c}:\mathcal{A}\to \mathcal{A}$$

given by

$${\mathcal{V}}_{\vartheta }^{a,c}f\left(z\right)=\frac{\Gamma (c+\vartheta )}{\Gamma (a+\vartheta )}\frac{1}{\Gamma (c-a)}\underset{0}{\overset{1}{\int }}{(1-t)}^{c-a-1}{t}^{a-1}f\left(z{t}^{\vartheta }\right)dt,\vartheta >0.$$

(3)

For $f\in \mathcal{A}$ assumed by (1), by simple computation, we have

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{\vartheta }^{a,c}f\left(z\right)& =& z+\sum _{n=2}^{\infty }\frac{\Gamma (c+\vartheta )\Gamma (a+n\vartheta )}{\Gamma (a+\vartheta )\Gamma (c+n\vartheta )}{a}_n{z}^n(z\in \Delta )\hfill \\ & =& z+\sum _{n=2}^{\infty }{{\rm Y}}_{\vartheta }^{a,c}\left(n\right)a_n{z}^n(z\in \Delta ),\hfill \end{array}$$

(4)

where

$${{\rm Y}}_{\vartheta }^{a,c}\left(n\right)=\frac{\Gamma (c+\vartheta )\Gamma (a+n\vartheta )}{\Gamma (a+\vartheta )\Gamma (c+n\vartheta )}.$$

By fixing $c=a$, we obtain

$${\mathcal{V}}_{\vartheta }^{a,a}f\left(z\right)=f\left(z\right).$$

(5)

Some applications and properties of the E–K fractional integral operator can be found in [2,3,4,5,6,7,10] and references therein. Now, we exploit the concept of the Sălăgean-Erdély–Kober fractional-order derivative(SEK) to define a new unified class of bi-univalent functions.

Definition 2.

Sălăgean-Erdély–Kober operator (SEK): For $m\in \mathbb{Z}=\{0,\pm 1,\pm 2,\pm 3,\cdots \};$$\vartheta >0,\Re (c-a)\ge 0,$$\Re \left(a\right)>-\vartheta ;\ell >-1;\varrho >0$ and $f\in \mathcal{A}$ is assumed by (1), we have

$$\begin{array}{ccc}\hfill {\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)& =& \frac{\Gamma (c+\vartheta )}{\Gamma (a+\vartheta )}\frac{1}{\Gamma (c-a)}\underset{0}{\overset{1}{\int }}{(1-t)}^{c-a-1}{t}^{a-1}{D}_{\ell ,\varrho }^{m}f\left(z{t}^{\vartheta }\right)dt,\vartheta >0\hfill \\ & =& z+\sum _{n=2}^{\infty }\frac{\Gamma (c+\vartheta )\Gamma (a+n\vartheta )}{\Gamma (a+\vartheta )\Gamma (c+n\vartheta )}{\left[1+\frac{\varrho (n-1)}{\ell +1}\right]}^m{a}_n{z}^n(z\in \Delta )\hfill \\ & =& z+\sum _{n=2}^{\infty }{{\rm Y}}_{\vartheta ,m,\ell }^{a,c,\varrho }{a}_n{z}^n,(z\in \Delta ),\hfill \end{array}$$

(6)

where

$${{\rm Y}}_n={{\rm Y}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(n\right)=\frac{\Gamma (c+\vartheta )\Gamma (a+n\vartheta )}{\Gamma (a+\vartheta )\Gamma (c+n\vartheta )}{\left[1+\frac{\varrho (n-1)}{\ell +1}\right]}^m.$$

(7)

Particularly,

$${{\rm Y}}_2={{\rm Y}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(2\right)={\left[1+\frac{\varrho }{\ell +1}\right]}^m\frac{\Gamma (c+\vartheta )\Gamma (a+2\vartheta )}{\Gamma (a+\vartheta )\Gamma (c+2\vartheta )}.$$

(8)

$${{\rm Y}}_3={{\rm Y}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(3\right)={\left[1+\frac{2\varrho }{\ell +1}\right]}^m\frac{\Gamma (c+\vartheta )\Gamma (a+3\vartheta )}{\Gamma (a+\vartheta )\Gamma (c+3\vartheta )}.$$

(9)

${\Xi }_{\vartheta ,m,\ell }^{a,a,\varrho }$ includes various differential and integral operators, as illustrated below:

Remark 1.

By fixing $m=0,$ operator ${\Xi }_{\vartheta ,m,\ell }^{a,a,\varrho }\equiv {\mathcal{V}}_{\vartheta }^{a,c}$, and suitably choosing parameters $a,c,\vartheta$ we obtain

1. 

For $a=\varsigma ;c=\sigma +\varsigma \vartheta =1,and$$\sigma \ge 0;\varrho >1$ we obtain the operator ${\mathcal{Q}}_{\varsigma }^{\sigma }$ studied by Jung et al. [11];

2. 

For $a=\sigma -1;c=\varsigma -1and\vartheta =1,$ with $\sigma ;\varsigma \in \mathbb{C}\in {\mathbb{Z}}_0;{\mathbb{Z}}_0=\{0;-1;-2;\cdots \}$ we obtain the operator ${\mathcal{L}}_{\sigma ,\varsigma }f\left(z\right)$ studied by Carlson and Shafer [12];

3. 

For $a=\rho -1;c=\ell and\vartheta =1,$ with $\rho >0;\ell >1$, we obtain the operator ${\mathcal{V}}_{\rho ,\ell }$ studied by Choi et al. [13];

4. 

For $a=\sigma ;c=0and\vartheta =1,$ with $\sigma >-1$ we obtain the operator ${\mathcal{D}}^{\sigma }$ studied by Ruscheweyh [14];

5. 

For $a=1;c=n\in {\mathbb{N}}_0=\{0,1,2,3,\cdots \}and\mu =1,$ we obtain the operator ${\mathcal{V}}_n$ studied in [15,16];

6. 

For $a=\varsigma ;c=\varsigma +1and\vartheta =1;$ we obtain the Bernardi integral operator [17]denoted as ${\mathcal{V}}_{\varsigma ,1}$;

7. 

For $a=1;c=2and\vartheta =1,$ give the Libera integral operator [18] as ${\mathcal{V}}_{1,1}=\mathcal{I}$ and Livingston [19].

Remark 2.

Let $m\in {\mathbb{N}}_0=\{0,1,2,3,\cdots .\}$ By fixing the values of $a,c,\vartheta$ as specified below, ${\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }$ includes various operators as cited below:

1. 

By fixing $\ell =0$ we have ${\Xi }_{\vartheta ,m,\ell }^{a,a,\varrho }\equiv {\mathcal{V}}_{\vartheta ,m}^{\varrho }$, Al-Oboudi operator [20].

2. 

Assuming $\varrho =1;\ell =0$ then ${\Xi }_{\vartheta ,m,\ell }^{a,a,\varrho }\equiv {\mathcal{V}}_{\vartheta }^m,$ Salagean operator [8].

3. 

Assuming $c=0;\vartheta =1$ then ${\Xi }_{1,m,\ell }^{a,0,\varrho }\equiv {\mathcal{V}}_{m,\varrho }^{a,\ell },$ Catas operator [21].

4. 

By fixing $\varrho =1;\ell =\eta$ and ${\Xi }_{\vartheta ,-m,\ell }^{a,a,\varrho }\equiv {\mathcal{V}}_{-m}^{\eta +1},$ Komatu operator [22].

Fractional calculus operators have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates, distortion inequalities, and convolutional structures of various subclasses of analytic functions. In this article, we study the subclasses of bi-univalent functions.

Bi-Univalent Functions $\Sigma$

The renowned Koebe one-quarter theorem (see [1]) asserts that the image of $\Delta$ under every univalent function $f\in \mathcal{A}$ contains a disk of radius $\frac{1}{4}$. Thus, the inverse of $f\in \mathcal{A}$ is a univalent analytic function on the disk ${\Delta }_{\rho }:=\{z:z\in \mathbb{C}\mathrm{and}|z|<\rho ;\rho \ge \frac{1}{4}\}$. Consequently, for each function $f\left(z\right)=w\in \sigma ,$ there is an inverse function $f^{-1}\left(w\right)$ of $f\left(z\right)$ defined by

$$f^{-1}\left(f\left(z\right)\right)=z(z\in \Delta ),$$

and

$$f\left(f^{-1}\left(w\right)\right)=w(w\in {\Delta }_{\rho }),$$

where

$$f^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+.\dots$$

(10)

A function $f\in \mathcal{A}$ is supposed to be bi-univalent in $\Delta$ if both f and $f^{-1}$ are univalent in $\Delta$.

Let $\Sigma$ denotes the class of bi-univalent functions in $\Delta$ given by (1). Note that the functions

$$f_1\left(z\right)={\displaystyle \frac{z}{1-z}},f_2\left(z\right)={\displaystyle \frac{1}{2}}log{\displaystyle \frac{1+z}{1-z}},f_3\left(z\right)=-log(1-z)$$

(11)

with their corresponding inverses

$$f_1^{-1}\left(w\right)={\displaystyle \frac{w}{1+w}},f_2^{-1}\left(w\right)={\displaystyle \frac{e^{2w}-1}{e^{2w}+1}},f_3^{-1}\left(w\right)={\displaystyle \frac{e^w-1}{e^w}}$$

(12)

are elements of $\Sigma .$ The concept of bi-univalent analytic functions was introduced by Lewin [23] in 1967, and he showed that $|a_2|<1.51.$ Subsequently, Brannan and Clunie [24] conjectured that $|a_2|\le \sqrt{2}.$ Netanyahu [25], on the other hand, showed that ${max}_{f\in \Sigma }|a_2|=\frac{4}{3}.$ The coefficient estimate problem for each of the following Taylor–Maclaurin coefficients:

$$|a_n|(n\in \mathbb{N}\setminus \{1,2\})$$

is presumably still an open problem. Recently, there has been interest in studying bi-univalent function class $\Sigma$ and obtained non-sharp coefficient estimates on the first two coefficients $|a_2|$ and $|a_3|$ of (1). This subject was extensively discussed in the pioneering work by Srivastava et al. [26], who revived the study of analytic and bi-univalent functions in recent years. It was followed by many sequels to Srivastava et al. [26] (see, for example, [27,28,29,30,31,32,33,34,35,36]), certain subclasses of the bi-univalent analytic functions class $\Sigma$ were introduced, and nonsharp estimates on the first two coefficients $|a_2|$ and $|a_3|$ were found. The study of operators plays an significant role in geometric function theory. Many differential and integral operators can be written in terms of the convolution of certain analytic functions. This formalism brings ease in further mathematical exploration, and helps in better understanding the symmetric and geometric properties of such operators. Inspired by the aforementioned works on bi-univalent functions, and by using Sălăgean-Erdély–Kober operator in the present paper, we define two new subclasses as in Definitions 3 and 4 of function class $\Sigma$, determine the estimates on coefficients $|a_2|$, $|a_3|$, and attempted to find $|a_4|$ for the functions in these new subclasses of function class $\Sigma$. We also discussed the Fekete-Szegö inequalities results [37] for $f\in {\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t),$ and $f\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t).$ Further we discussed the results several consequences of the results for the new subclasses of $\Sigma$ that are not studied in association with telephone numbers based on Sălăgean-Erdély–Kober operator as illustrated in Definitions 5–8.

Definition 3.

For $0<\xi \le 1,t\ge 1,$ and $f\in \Sigma$ be assumed by (1) is supposed to be in class ${\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t)$ if the following conditions are satisfied:

$$\left|\arg \left((1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right)}^{\prime }\right)\right|<\frac{\xi \pi }{2},$$

(13)

and

$$\left|\arg \left((1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right)}^{\prime }\right)\right|<\frac{\xi \pi }{2},$$

(14)

where $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g are the inverse of f given by (10).

Definition 4.

For $0<\zeta \le 1,t\ge 1;$ and $f\in \Sigma$ are given by (1); then, $f\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t)$ if the following conditions are satisfied:

$$\Re \left((1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right)}^{\prime }\right)>\zeta$$

(15)

and

$$\Re \left((1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right)}^{\prime }\right)>\zeta ,$$

(16)

where $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g is the inverse of f given by (10).

By fixing $t=1,$ we define a new subclass of $\Sigma$ due to Noshiro [38].

Definition 5.

For $0<\xi \le 1$ and $f\in \Sigma$ be given by (1) then $f\in {\mathcal{N}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(\xi \right)$ if it holds the following conditions:

$$\left|\arg \left({\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right)}^{\prime }\right)\right|<\frac{\xi \pi }{2}\mathrm{a}\mathrm{nd}\left|\arg \left({\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right)}^{\prime }\right)\right|<\frac{\xi \pi }{2},$$

(17)

where $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g is the inverse of f given by (10).

Definition 6.

A function $f\in \Sigma$ given by (1) then $f\in {\mathcal{M}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(\zeta \right)$ if it satisfy the following conditions:

$$\Re \left({\left[{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right]}^{\prime }\right)>\zeta \mathrm{a}\mathrm{nd}\Re \left({\left[{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right]}^{\prime }\right)>\zeta ,$$

(18)

where $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g is the inverse of f given by (10).

By fixing $t=0,$ we define a new subclass of $\Sigma$ due to Yamaguchi [39].

Definition 7.

For $0<\xi \le 1$ and $f\in \Sigma$, as assumed as (1); then, $f\in {\mathcal{Y}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(\xi \right)$ if the following conditions hold:

$$\left|\arg \left(\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}\right)\right|<\frac{\xi \pi }{2}\mathrm{a}\mathrm{nd}\left|\arg \left(\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}\right)\right|<\frac{\xi \pi }{2},$$

(19)

where $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g are the inverse of f given by (10).

Definition 8.

For $0\le \zeta <1$ and a function $f\in \Sigma$ given by (1); then, $f\in {\mathcal{X}}_{\vartheta ,m,\ell }^{a,c,\varrho }\left(\zeta \right)$ if the following conditions are satisfied:

$$\Re \left(\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}\right)>\zeta \mathrm{a}\mathrm{nd}\Re \left(\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}\right)>\zeta ,$$

(20)

where $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g are the inverse of f given by (10).

2. Coefficient Bounds

In order to find the initial coefficient bounds, namely, $|a_2|,|a_3|$ and $|a_4|$ for $f\in {\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t)$ and $f\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t)$ of $\Sigma ,$ we recall the following lemma:

Lemma 1.

(see [1], p. 41) Let $\varphi \left(z\right)=1+{\sum }_{n=1}^{\infty }{c}_n{z}^n$ be in $\mathcal{P},$ the class of all analytic functions with $\Re \left(\varphi \right(z\left)\right)>0(z\in \Delta )$ and $\varphi \left(0\right)=1.$ Then,

$$|c_n|\le 2(n=1,2,3,\dots ).$$

This inequality is sharp $\forall n$. In particular, for $\varphi \left(z\right)=\frac{1+z}{1-z}=1+{\sum }_{n=1}^{\infty }2z^n.$ equality holds $\forall n$.

Theorem 1.

Let $f\left(z\right)$ given by (1) be in the class ${\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t)$. Then

$$|a_2|\le \frac{2\xi }{\sqrt{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}},$$

(21)

$$|a_3|\le \frac{2\xi }{(1+2t){{\rm Y}}_3},$$

(22)

and

$$|a_4|\le \frac{2\xi }{(1+3t){{\rm Y}}_4}\left[1+\frac{2(1-\xi )(1+t){{\rm Y}}_2\{6\xi (1+2t){{\rm Y}}_3+(1-2\xi ){(1+t)}^2{{\rm Y}}_2^2\}}{3{\{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2\}}^{\frac{3}{2}}}\right].$$

(23)

Proof. 

Let $f\in {\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t)$. Hence, by Definition 3, there exists $\varphi \left(z\right)\mathrm{and}\psi \left(w\right)\in \mathcal{P}$ such that

$$(1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right)}^{\prime }={\left[\varphi \left(z\right)\right]}^{\xi }$$

(24)

and

$$(1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right)}^{\prime }={\left[\psi \left(w\right)\right]}^{\xi }.$$

(25)

Write

$$\varphi \left(z\right)=1+c_{1}z+c_2{z}^2+c_3{z}^3+\dots$$

(26)

and

$$\psi \left(w\right)=1+d_{1}w+d_2{w}^2+d_3{w}^3+.\dots ,$$

(27)

Now, equating the coefficients in (24) and (25), we obtain

$$(1+t){{\rm Y}}_2{a}_2=\xi c_1$$

(28)

$$(1+2t){{\rm Y}}_3{a}_3=\xi c_2+\frac{\xi (\xi -1)}{2}{c}_1^2$$

(29)

$$(1+3t){{\rm Y}}_4{a}_4=\xi c_3+\xi (\xi -1)c_1{c}_2+\frac{\xi (\xi -1)(\xi -2)}{6}{c}_1^3,$$

(30)

and

$$-(1+t){{\rm Y}}_2{a}_2=\xi d_1$$

(31)

$$(1+2t){{\rm Y}}_3(2a_2^2-a_3)=\xi d_2+\frac{\xi (\xi -1)}{2}{d}_1^2$$

(32)

$$-(1+3t){{\rm Y}}_4(5a_2^3-5a_2{a}_3+a_4)=\xi d_3+\xi (\xi -1)d_1{d}_2+\frac{\xi (\xi -1)(\xi -2)}{6}{d}_1^3.$$

(33)

From (28) and (31), we obtain

$$a_2=\frac{\xi c_1}{(1+t){{\rm Y}}_2}=-\frac{\xi }{(1+t){{\rm Y}}_2}{d}_1$$

(34)

which implies

$$c_1=-d_1.$$

Squaring and adding (28), (31), we obtain

$$a_2^2=\frac{{\xi }^2}{{(1+t)}^2{{\rm Y}}_2^2}(c_1^2+d_1^2).$$

(35)

Adding (29) and (32), we obtain

$$2(1+2t){{\rm Y}}_3{a}_2^2=\xi (c_2+d_2)+\frac{\xi (\xi -1)}{2}(c_1^2+d_1^2).$$

Substitute the value of $a_2$ from (34) in (35) and noting that $c_1^2=d_1^2,$ we observe that

$$c_1^2=\frac{{(1+t)}^2{{\rm Y}}_2^2(c_2+d_2)}{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}.$$

(36)

By applications of the triangle inequality and Lemma 1 give

$$|c_1|\le \frac{2(1+t){{\rm Y}}_2}{\sqrt{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}}.$$

(37)

Thus, (34) gives

$$|a_2|\le \frac{2\xi }{\sqrt{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}}.$$

(38)

In order to find the bound on $|a_3|$, subtracting (32) from (29) with $c_1=-d_1$ gives

$$\begin{array}{ccc}\hfill 2(1+2t){{\rm Y}}_3{a}_3& =& 2(1+2t){{\rm Y}}_3{a}_2^2+\xi (c_2-d_2)\hfill \\ \hfill a_3& =& a_2^2+\frac{\xi (c_2-d_2)}{2(1+2t){{\rm Y}}_3}.\hfill \end{array}$$

(39)

Using (34) and (36) in (39) after simplification yields

$$\begin{array}{ccc}\hfill 2(1+2t){{\rm Y}}_3{a}_3& =& \frac{2{\xi }^2{{\rm Y}}_3(1+2t)}{2\xi (1+2t){{\rm Y}}_3+{(1+t)}^2{{\rm Y}}_2^2(1-\xi )}(c_2+d_2)+\xi (c_2-d_2)\hfill \\ & =& \left(\frac{2{\xi }^2{{\rm Y}}_3(1+2t)}{2\xi (1+2t){{\rm Y}}_3+{(1+t)}^2{{\rm Y}}_2^2(1-\xi )}+\xi \right)c_2\hfill \\ & +& \left(\frac{2{\xi }^2{{\rm Y}}_3(1+2t)}{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}-\xi \right)d_2\hfill \\ & =& \frac{\xi \left[\left\{4\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2\right\}{c}_2-(1-\xi ){(1+t)}^2{{\rm Y}}_2^2{d}_2\right]}{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}\hfill \end{array}$$

By the application of a triangle inequality to the above equation,

$$|a_3|\le \frac{\xi \left[\left\{4\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2\right\}|c_2{|+(1-\xi )(1+t)}^2{{\rm Y}}_2^2|d_2|\right]}{2(1+2t){{\rm Y}}_3\left[2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2\right]}.$$

Again, by applying Lemma 1 for the coefficients $c_2$ and $d_2$, we obtain

$$|a_3|\le \frac{2\xi }{(1+2t){{\rm Y}}_3}.$$

To determine the bound on $|a_4|$, adding (30) and (33) with $c_1=-d_1$, we have

$$-5(1+3t){{\rm Y}}_4{a}_2^3+5(1+3t){{\rm Y}}_4{a}_2{a}_3=\xi (c_3+d_3)+\xi (\xi -1)c_1(c_2-d_2),$$

(40)

substitute the values of $a_2$ and $a_3$ from (34) and (39) in (40) and simplify, we obtain

$$c_1(c_2-d_2)=\frac{2(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}{5\xi (1+3t){{\rm Y}}_4+2(1-\xi )(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}(c_3+d_3).$$

(41)

Subtracting (33) from (30) and using (36), (37), (40) and (41) in the result, we obtain

$$\begin{array}{ccc}\hfill 2a_4(1+3t){{\rm Y}}_4& =\hfill & -5(1+3t){{\rm Y}}_4{a}_2^3+5(1+3t)a_2{a}_3{{\rm Y}}_4+\xi (c_3-d_3)\hfill \\ \hfill & +\hfill & \xi (\xi -1)c_1(c_2+d_2)+\frac{\xi (\xi -1)(\xi -2)}{3}{c}_1^3\hfill \\ \hfill & =\hfill & \xi (c_3+d_3)+\xi (\xi -1)c_1(c_2-d_2)\hfill \\ \hfill & +\hfill & \xi (c_3-d_3)+\xi (\xi -1)c_1(c_2+d_2)+\frac{\xi (\xi -1)(\xi -2)}{3}{c}_1^3\hfill \\ \hfill & =\hfill & 2\xi c_3+\frac{2\xi (\xi -1)(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}{5\xi (1+3t){{\rm Y}}_4+2(1-\xi )(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}(c_3+d_3)\hfill \\ \hfill & +\hfill & \xi (\xi -1)c_1(c_2+d_2)+\frac{\xi (\xi -1)(\xi -2){(1+t)}^2{{\rm Y}}_2^2}{3\{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2\}}\hfill \\ \hfill & =\hfill & \frac{10{\xi }^2(1+3t){{\rm Y}}_4+2\xi (1-\xi )(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}{5\xi (1+3t){{\rm Y}}_4+2(1-\xi )(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}{c}_3\hfill \\ \hfill & -\hfill & \frac{2\xi (1-\xi )(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}{5(1+3t)\xi {{\rm Y}}_4+2(1-\xi )(1+2t)(1+t){{\rm Y}}_2{{\rm Y}}_3}{d}_3\hfill \\ \hfill & -\hfill & \xi (1-\xi )\left[\frac{6\xi (1+2t){{\rm Y}}_3+(1-2\xi ){(1+t)}^2{{\rm Y}}_2^2}{6\xi (1+2t){{\rm Y}}_3+3(1-\xi ){(1+t)}^2{{\rm Y}}_2^2}\right]c_1(c_2+d_2).\hfill \end{array}$$

(42)

Applying Lemma 1 with the triangle inequality in (42), we obtain

$$|a_4|\le \frac{2\xi }{(1+3t){{\rm Y}}_4}\left[1+\frac{2(1-\xi )(1+t){{\rm Y}}_2\left\{6\xi (1+2t){{\rm Y}}_3+(1-2\xi ){(1+t)}^2{{\rm Y}}_2^2\right\}}{3{\left\{2\xi (1+2t){{\rm Y}}_3+(1-\xi ){(1+t)}^2{{\rm Y}}_2^2\right\}}^{\frac{3}{2}}}\right].$$

This concludes the proof of Theorem 1. □

Theorem 2.

Let $f\left(z\right)$ be given by (1) and $f\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t)$. Then

$$|a_2|\le \sqrt{\frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3}},$$

(43)

$$|a_3|\le \frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3}$$

(44)

and

$$|a_4|\le \frac{2(1-\zeta )}{(1+3t){{\rm Y}}_4}.$$

(45)

Proof. 

Since $f\left(z\right)\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t)$, there exist two functions $\varphi \left(z\right)$ and $\psi \left(z\right)\in \mathcal{P}$ satisfying the conditions of Lemma 1, such that

$$(1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right)}^{\prime }=\zeta +(1-\zeta )\varphi \left(z\right)$$

(46)

and

$$(1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right)}^{\prime }=\zeta +(1-\zeta )\psi \left(w\right)$$

(47)

where $\varphi \left(z\right)$ and $\psi \left(w\right)$ given by (26) and (27) respectively. Equating the coefficients in (46) and (47) gives

$$(1+t){{\rm Y}}_2{a}_2=(1-\zeta )c_1$$

(48)

$$(1+2t){{\rm Y}}_3{a}_3=(1-\zeta )c_2$$

(49)

$$(1+3t){{\rm Y}}_4{a}_4=(1-\zeta )c_3,$$

(50)

and

$$-(1+t){{\rm Y}}_2{a}_2=(1-\zeta )d_1$$

(51)

$$(1+2t){{\rm Y}}_3(2a_2^2-a_3)=(1-\zeta )d_2$$

(52)

$$-(1+3t){{\rm Y}}_4(5a_2^3-5a_2{a}_3+a_4)=(1-\zeta )d_3,$$

(53)

from (48) and (51) gives

$$a_2=\frac{1-\zeta }{(1+t){{\rm Y}}_2}{c}_1=-\frac{1-\zeta }{(1+t){{\rm Y}}_2}{d}_1,$$

(54)

which implies

$$c_1=-d_1.$$

Adding (49) and (52), we obtain

$$\begin{array}{ccc}\hfill 2(1+2t){{\rm Y}}_3{a}_2^2& =& (1-\zeta )(c_2+d_2)\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill a_2^2& =& \frac{(1-\zeta )}{2(1+2t){{\rm Y}}_3}(c_2+d_2),\hfill \end{array}$$

(56)

using (54) in (55), we have

$$c_1^2=\frac{{(1+t)}^2{{\rm Y}}_2^2}{2(1+2t)(1-\zeta ){{\rm Y}}_3}(c_2+d_2),$$

(57)

an applications of the triangle inequality and Lemma 1 in (57) yield

$$|c_1|\le (1+t){{\rm Y}}_2\sqrt{\frac{2}{(1+2t)(1-\zeta ){{\rm Y}}_3}},$$

(58)

using (58) in (54) gives

$$|a_2|\le \sqrt{\frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3}}.$$

(59)

Now, subtracting (52) from (49) and using (55), we obtain

$$|a_3|\le \frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3},$$

(60)

which is the direct consequence of (49).

In order to obtain the bounds on $|a_4|$, we proceed as follows:

From (50), it is easy to see that

$$|a_4|=\left|\frac{(1-\zeta )c_3}{(1+3t){{\rm Y}}_4}\right|\le \frac{2(1-\zeta )}{(1+3t){{\rm Y}}_4}.$$

(61)

On the otherhand, subtracting (53) from (50) and using (54), we obtain

$$a_4=\frac{1}{2(1+3t){{\rm Y}}_4}\left[\frac{-5(1+3t){(1-\zeta )}^3{{\rm Y}}_4}{{(1+t)}^3{{\rm Y}}_2^3}{c}_1^3+\frac{5(1+3t)(1-\zeta ){{\rm Y}}_4}{(1+t){{\rm Y}}_2}{a}_3{c}_1+(1-\zeta )(c_3-d_3)\right].$$

(62)

Applying the triangle inequality in (62), we have

$$|a_4|\le \frac{1}{2(1+3t){{\rm Y}}_4}\left[\frac{5(1+3t){(1-\zeta )}^3{{\rm Y}}_4}{{(1+t)}^3{{\rm Y}}_2^3}|c_1{|}^3+\frac{5(1+3t)(1-\zeta ){{\rm Y}}_4}{(1+t){{\rm Y}}_2}|a_3||c_1|+(1-\zeta )(|c_3|+|d_3|)\right].$$

(63)

Using (58), (60) and Lemma 1 in (63), after simplification yield

$$|a_4|\le \frac{2(1-\zeta )}{(1+3t){{\rm Y}}_4}\left[1+\frac{5(1+3t)}{(1+2t){{\rm Y}}_3}\sqrt{\frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3}}\right].$$

(64)

From (61) and (64), we observe that

$$\begin{array}{cc}\hfill |a_4|& \le min\left[\frac{2(1-\zeta )}{(1+3t){{\rm Y}}_4},\frac{2(1-\zeta )}{(1+3t){{\rm Y}}_4}\left\{1+\frac{5(1+3t)}{(1+2t){{\rm Y}}_3}\sqrt{\frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3}}\right\}\right]\hfill \\ \hfill & =\frac{2(1-\zeta )}{(1+3t){{\rm Y}}_4}.\hfill \end{array}$$

This completes the proof of Theorem 2. □

3. Fekete-Szegö Inequalities

In this section, we obtain Fekete-Szegö inequalities results [37] (see [40]), for $f\in {\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t),$ and $f\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t).$

Theorem 3.

For $\nu \in \mathbb{R},$ let f be given by (1) and $f\in {\mathcal{YN}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\xi ,t),$ then

$$\left|a_3-\nu a_2^2\right|\le \left\{\begin{array}{cc}\frac{2\xi }{(1+2t){{\rm Y}}_3},\hfill & 0\le \left|h\left(\nu \right)\right|\le \frac{\xi }{2(1+2t){{\rm Y}}_3};\hfill \\ 4\left|h\left(\nu \right)\right|,\hfill & \left|h\left(\nu \right)\right|\ge \frac{\xi }{2(1+2t){{\rm Y}}_3},\hfill \end{array}\right.$$

where

$$\begin{array}{c}\hfill h\left(\nu \right)=\frac{2{\xi }^2(1-\nu )}{4\xi (1+2t){{\rm Y}}_3-(\xi -1){(1+t)}^2{{\rm Y}}_2^2}.\end{array}$$

Proof. 

From (39), we have

$$\begin{array}{ccc}\hfill a_3-\nu a_2^2& =& \frac{\xi (c_2-d_2)}{2(1+2t){{\rm Y}}_3}+(1-\nu )a_2^2\hfill \end{array}$$

(65)

$$\begin{array}{ccc}& =& \frac{\xi (c_2-d_2)}{2(1+2t){{\rm Y}}_3}+\frac{2{\xi }^2(1-\nu )(c_2+d_2)}{4\xi (1+2t){{\rm Y}}_3-(\xi -1){(1+t)}^2{{\rm Y}}_2^2}.\hfill \end{array}$$

(66)

By simple computation, we have

$$a_3-\nu a_2^2=\left(h\left(\nu \right)+\frac{\xi }{2(1+2t){{\rm Y}}_3}\right)c_2+\left(h\left(\nu \right)-\frac{\xi }{2(1+2t){{\rm Y}}_3}\right)d_2,$$

(67)

where

$$h\left(\nu \right)=\frac{2{\xi }^2(1-\nu )}{4\xi (1+2t){{\rm Y}}_3-(\xi -1){(1+t)}^2{{\rm Y}}_2^2}.$$

(68)

Thus by taking modulus of (67), we obtain

$$\left|a_3-\nu a_2^2\right|\le \left\{\begin{array}{cc}\frac{2\xi }{(1+2t){{\rm Y}}_3}\hfill & ;0\le \left|h\left(\nu \right)\right|\le \frac{\xi }{2(1+2t){{\rm Y}}_3}\hfill \\ 4\left|h\left(\nu \right)\right|\hfill & ;\left|h\left(\nu \right)\right|\ge \frac{\xi }{2(1+2t){{\rm Y}}_3},\hfill \end{array}\right.$$

where $h\left(\nu \right)$ is given by (68). □

Theorem 4.

For $\nu \in \mathbb{R},$ let f be given by (1) and $f\in {\mathcal{YM}}_{\vartheta ,m,\ell }^{a,c,\varrho }(\zeta ,t),$ then

$$\left|a_3-\nu a_2^2\right|\le \frac{(1-\zeta )|2-\nu |}{(1+2t){{\rm Y}}_3}\left[1+\frac{\nu }{|2-\nu |}\right].$$

Proof. 

Subtracting (52) from (49), we have

$$a_3=\frac{(1-\zeta )(c_2-d_2)}{2(1+2t){{\rm Y}}_3}+a_2^2,$$

(69)

and using (56), we obtain

$$\begin{array}{ccc}\hfill a_3-\nu a_2^2& =& \frac{(1-\zeta )(c_2-d_2)}{2(1+2t){{\rm Y}}_3}+(1-\nu )a_2^2\hfill \end{array}$$

(70)

$$\begin{array}{ccc}& =& \frac{(1-\zeta )(c_2-d_2)}{2(1+2t){{\rm Y}}_3}+\frac{(1-\zeta )(1-\nu )}{2(1+2t){{\rm Y}}_3}(c_2+d_2).\hfill \end{array}$$

(71)

By simple computation, we have

$$\begin{array}{ccc}\hfill a_3-\nu a_2^2& =\hfill & \left(\frac{(1-\zeta )(1-\nu )}{2(1+2t){{\rm Y}}_3}+\frac{1-\zeta }{2(1+2t){{\rm Y}}_3}\right)c_2+\left(\frac{(1-\zeta )(1-\nu )}{2(1+2t){{\rm Y}}_3}-\frac{1-\zeta }{2(1+2t){{\rm Y}}_3}\right)d_2,\hfill \\ \hfill & =\hfill & \frac{(1-\zeta )}{2(1+2t){{\rm Y}}_3}\left((1-\nu )+1\right)c_2+\frac{(1-\zeta )}{2(1+2t){{\rm Y}}_3}\left((1-\nu )-1\right)d_2,\hfill \\ \hfill & =\hfill & \frac{(1-\zeta )}{2(1+2t){{\rm Y}}_3}\left[(2-\nu )c_2-\nu d_2\right],\hfill \\ \hfill & =\hfill & \frac{(1-\zeta )(2-\nu )}{2(1+2t){{\rm Y}}_3}\left[c_2-\frac{\nu }{2-\nu }{d}_2\right].\hfill \end{array}$$

(72)

Thus, by taking the modulus of (72), we obtain

$$\left|a_3-\nu a_2^2\right|\le \frac{(1-\zeta )|2-\nu |}{(1+2t){{\rm Y}}_3}\left[1+\frac{\nu }{|2-\nu |}\right].$$

In particular $\nu =1,$ we obtain

$$\left|a_3-a_2^2\right|\le \frac{2(1-\zeta )}{(1+2t){{\rm Y}}_3}.$$

□

4. Conclusions

We defined a unified Yamaguchi–Noshiro type subclass of bi-univalent functions based on Sălăgean-Erdély–Kober operator. We obtained nonsharp bounds for the initial coefficients and the Fekete–Szegö inequalities for the functions in this new class. Some interesting corollaries and applications of the results were also discussed. By suitably fixing the parameters, as illustrated in Remarks 1 and 1, one can easily state the results discussed in this article for the function classes given in Definitions 3–8. One can construct the Yamaguchi–Noshiro class on the basis of the Ma–Minda [41] subordination [42] for a given $f\in \Sigma$ to be given by (1), satisfying the following conditions:

$$\left((1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)}{z}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }f\left(z\right)\right)}^{\prime }\right)\prec \phi \left(z\right)$$

(73)

and

$$\left((1-t)\frac{{\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)}{w}+t{\left({\Xi }_{\vartheta ,m,\ell }^{a,c,\varrho }g\left(w\right)\right)}^{\prime }\right)\prec \phi \left(w\right),$$

(74)

where

$$\phi \left(z\right)=1+B_{1}z+B_2{z}^2+B_3{z}^3+\cdots ,(B_1>0)$$

$0<\zeta \le 1,t\ge 1,$ and $z,\in \Delta ;w\in {\Delta }_{\rho }$ and g are the inverse of f given by (10).

By giving some specific values to $\phi$, as listed below, we define several new subclasses of $\Sigma$:

• For $\phi \left(z\right)=1+sinz,$ we have the function class ${\mathcal{S}}^{\ast }\left(\phi \right)$ of starlike functions associated with the sine functions (see [43]).
• For $\phi \left(z\right)=1+z-\frac{1}{3}{z}^3,$ we have the function class of starlike functions associated with the nephroid (see [44]).
• For $\phi \left(z\right)=\sqrt{1+z},$ we have the function class of starlike functions associated with the lemniscate of Bernoulli (see [45]).
• For $\phi \left(z\right)=e^z,$ we have the function class of starlike functions associated with the exponintial functions (see [46]).
• For $\phi \left(z\right)=z+\sqrt{1+z^2},$ we have the function class of starlike functions associated with the crescent shaped region (see [47]).