A Family of Analytic and Bi-Univalent Functions Associated with Srivastava-Attiya Operator

Abstract

In this paper, we investigate a new family of normalized analytic functions and bi-univalent functions associated with the Srivastava–Attiya operator. We use the Faber polynomial expansion to estimate the bounds for the general coefficients $|a_n|$ of this family. The bounds values for the initial Taylor–Maclaurin coefficients of the functions in this family are also established.

1. Introduction

Let A denote the class of all normalized analytic functions of the form

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

(1)

which are defined in the open unit disk $\mathbb{U}=\{\zeta :\zeta \in \mathbb{C}$ and $\left|\zeta \right|<1\}$. Additionally, S denotes to the class of all functions in A which are univalent in $\mathbb{U}$.

For the function $f\in A,$ Airault and Bouali ([1], page 184) used Faber polynomial to show that

$$\frac{\zeta f^{{}^{\prime }}\left(\zeta \right)}{f\left(\zeta \right)}=1-{\displaystyle \sum _{j=2}^{\infty }}{F}_{j-1}(a_2,a_3,\dots ,a_j){\zeta }^{j-1},$$

(2)

where $F_{j-1}(a_2,a_3,\dots ,a_j)$ is the Faber polynomial defined by

$$F_{j-1}(a_2,a_3,\dots ,a_j)=\sum _{i_1+2i_2+\dots +(j-1)i_{j-1}=j-1}A(i_1,i_2,\dots ,i_{j-1})a_2^{i_1}{a}_3^{i_2}\dots ,a_j^{i_{j-1}}$$

(3)

and

$$A(i_1,i_2,\dots ,i_{j-1}):={(-1)}^{(j-1)+2i_1+\dots +j{i}_{j-1}}\frac{(i_1+i_2+\dots +i_{j-1}-1)!(j-1)}{\left(i_1\right)!\left(i_2\right)!\dots \left(i_{j-1}\right)!}.$$

The first terms of the Faber polynomial $F_{j-1},j\ge 2,$ are given by (e.g., see ([2], page 52)).

$$\begin{array}{cc}\mathrm{(4)}& \hfill F_1& =-a_2,\hfill \hfill F_2& =a_2^2-2a_3,\hfill \\ \hfill F_3& =-a_2^3+3a_2{a}_3-3a_4,\hfill \\ \hfill F_4& =a_2^4-4a_2^2{a}_3+4a_2{a}_4+2a_3^2-4a_5\hfill \\ \hfill F_5& =-a_2^5+5a_2^3{a}_3+5a_2^2{a}_4-5a_2(a_3^2-a_5)+5a_3{a}_4-5a_6.\hfill \end{array}$$

The theorem of Koebe one-quarter ([3], page 31) guarantees that the range of each function $f\in S$ contains the open disk with radius ${\displaystyle \frac{1}{4}}$. Therefore, each function $f\in S$ has an inverse $f^{-1},$ which is defined as follows:

$$f^{-1}\left(f\left(\zeta \right)\right)=\zeta (\zeta \in \mathbb{U})$$

and

$$f\left(f^{-1}\left(\omega \right)\right)=\omega (\left|\omega \right|<\frac{1}{4}).$$

The inverse function $g:=f^{-1}$ for each f$\in S$ has Taylor series expansion as follows (see ([1], page 185)):

$$\begin{array}{cc}\hfill g\left(\omega \right)& =f^{-1}\left(\omega \right)=w+{\displaystyle \sum _{n=2}^{\infty }}\frac{1}{n}{K}_{n-1}^{-n}(a_2,a_3,\dots ,a_n){\omega }^n\hfill \\ \hfill & =w-a_2{\omega }^2+(2a_2^2-a_3){\omega }^3-(5a_2^2-5a_2{a}_3+a_4){\omega }^4+\dots ,\hfill \end{array}$$

(5)

where the coefficients of $n$ parametric function $K_n^p(a_2,a_3,\dots ,a_n)$ are given by

$$\begin{array}{cc}\hfill K_1^p& =p{a}_2,\hfill \\ \mathrm{(6)}& \hfill K_2^p& =\frac{p(p-1)}{2}{a}_2^2+p{a}_3,\hfill \hfill K_3^p& =p(p-1)a_2{a}_3+p{a}_4+\frac{p(p-1)(p-2)}{3!}{a}_2^3,\hfill \\ \hfill K_4^p& =p(p-1)a_2{a}_4+p{a}_5+\frac{p(p-1)}{2}{a}_3^2+\frac{p(p-1)(p-2)}{2}{a}_2^2{a}_3+\frac{p!}{(p-4)!4!}{a}_2^4,\hfill \\ \hfill & ⋮\hfill \\ \hfill K_n^p& =\frac{p!}{(p-n)!n!}{a}_2^n+\frac{p!}{(p-n+1)!(n-2)!}{a}_2^{n-2}{a}_3+\frac{p!}{(p-n+2)!(n-3)!}{a}_2^{n-3}{a}_4\hfill \\ \mathrm{(7)}& \hfill & +\frac{p!}{(p-n+3)!(n-4)!}{a}_2^{n-4}\left[a_5+\frac{p-n+3}{2}{a}_3^2\right]\hfill \hfill & +\frac{p!}{(p-n+4)!(n-5)!}{a}_2^{n-4}\left[a_6+(p-n+3)a_3{a}_4\right]+{\displaystyle \sum _{j\ge 6}^{\infty }}{a}_2^{n-j}{V}_j,\hfill \end{array}$$

where $V_j$ is a homogeneous polynomial of jth degree in the variables $a_3,\dots ,a_n,$ see (([4], page 349) and ([1], pages 183 and 205)).

Lemma 1.

(Schwarz lemma ([3], page 3)) Let $\omega \left(\zeta \right)$ be analytic in $\mathbb{U},$ with $\omega \left(0\right)=0$ and $\left|\omega \left(\zeta \right)\right|<1$ in $\mathbb{U}$, then we have $\left|\omega \left(\zeta \right)\right|<\left|\zeta \right|$ and $\left|{\omega }^{{}^{\prime }}\left(0\right)\right|<1$ in $\mathbb{U}$.

If f and g are analytic functions in the open unit disk $\mathbb{U}$, then the function f is called subordinate to g, denoted by

$$f\prec g\mathrm{or}f\left(\zeta \right)\prec g\left(\zeta \right),$$

if there exists a Schwarz function w, which is analytic in $\mathbb{U}$ with $w\left(0\right)=0$ and $\left|w\left(\zeta \right)\right|<1$$(\zeta \in \mathbb{U})$, such that $f\left(\zeta \right)=g\left(w\left(\zeta \right)\right)$ for all $\zeta \in \mathbb{U}$. In particular, if the function g$\in S$ in $\mathbb{U}$, then we have the following equivalence relation (cf., e.g., [5,6]; see also [3]):

$$f\left(z\right)\prec g\left(z\right)\iff f\left(0\right)\prec g\left(0\right)andf\left(\mathbb{U}\right)\subset g\left(\mathbb{U}\right).$$

Let $\varphi$ be analytic function with $Re\left(\varphi \right)>0$ in $\mathbb{U}$, satisfies $\varphi \left(0\right)=1,{\varphi }^{{}^{\prime }}\left(0\right)>0,$ and $\varphi \left(\mathbb{U}\right)$ is symmetric with respect to the real axis. This function has a Taylor series expansion as follows

$$\varphi \left(\zeta \right)=1+B_1\zeta +B_2{\zeta }^2+B_3{\zeta }^3+\dots (B_1>0).$$

(8)

Srivastava and Attiya [7] introduced the operator $J_{s,b}\left(f\right):A\to A$ which makes a connection between Geometric Function Theory and Analytic Number Theory, defined by

$$J_{s,b}\left(f\right)\left(\zeta \right)=G_{s,b}\left(\zeta \right)\ast f\left(\zeta \right)$$

(9)

$$\left(\zeta \in \mathbb{U};f\in A;b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};s\in \mathbb{C}\right)$$

where

$$G_{s,b}\left(\zeta \right)={(1+b)}^s\left[\mathsf{\Phi }(\zeta ,s,b)-b^{-s}\right]$$

(10)

and ∗ denotes the Hadamard product (or Convolution) and a general Hurwitz–Lerch Zeta function $\mathsf{\Phi }(\zeta ,s,b)$ defined by (cf., e.g., ([8], P. 121 et seq.))

$$\mathsf{\Phi }(\zeta ,s,b)={\displaystyle \sum _{n=0}^{\infty }}\frac{{\zeta }^k}{{(n+b)}^s},$$

(11)

$(b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-},{\mathbb{Z}}_0^{-}={\mathbb{Z}}^{-}\cup \left\{0\right\}=\{0,-1,-2,\dots \},s\in \mathbb{C}when\zeta \in \mathbb{U},Re\left(s\right)>1when$$\left|\zeta \right|=1)$

Furthermore, Srivastava and Attiya [7] showed that

$$J_{s,b}\left(f\right)\left(\zeta \right)=\zeta +{\displaystyle \sum _{n=2}^{\infty }}{\left(\frac{1+b}{n+b}\right)}^s{a}_k{\zeta }^k.$$

(12)

Additionally, Srivastava and Attiya [7] deduced that, if $\zeta \in \mathbb{U},f\in A,b\in \mathbb{C}\setminus {\mathbb{Z}}^{-}$ and $s\in \mathbb{C},$ then

$$\zeta J_{s+1,b}^{{}^{\prime }}\left(f\right)\left(\zeta \right)=(1+b)J_{s,b}\left(f\right)\left(\zeta \right)-b{J}_{s+1,b}\left(f\right)\left(\zeta \right).$$

(13)

Noting that:

$$J_{0,b}\left(f\right)\left(\zeta \right)=f\left(\zeta \right),$$

(14)

$$J_{1,0}\left(f\right)\left(\zeta \right)=\underset{0}{\overset{\zeta }{\int }}\frac{f\left(t\right)}{t}dt=A\left(f\right)\left(\zeta \right),$$

(15)

$$J_{1,1}\left(f\right)\left(\zeta \right)=\frac{2}{\zeta }\underset{0}{\overset{\zeta }{\int }}f\left(t\right)dt=L\left(f\right)\left(\zeta \right),$$

(16)

$$J_{1,\gamma }\left(f\right)\left(\zeta \right)=\frac{1+\gamma }{{\zeta }^{\gamma }}\underset{0}{\overset{\zeta }{\int }}f\left(t\right)t^{\gamma -1}dt=L_{\gamma }\left(f\right)\left(\zeta \right)(\gamma \mathrm{real};\gamma >-1),$$

(17)

$$J_{\sigma ,1}\left(f\right)\left(\zeta \right)=\zeta +{\displaystyle \sum _{n=2}^{\infty }}{\left(\frac{2}{k+1}\right)}^{\sigma }{a}_n{\zeta }^k=I^{\sigma }\left(f\right)\left(\zeta \right)(\sigma \mathrm{real};\sigma >0),$$

(18)

for $f\left(\zeta \right)\in A,$$t_1;t_2;\dots ;t_n;\zeta \in \mathbb{U},n\in \mathbb{N}andb\in \mathbb{C}\setminus {\mathbb{Z}}^{-},$ we have

$$J_{n,b}\left(f\right)\left(\zeta \right)=\frac{{(1+b)}^n}{{\zeta }^b}\underset{0}{\overset{\zeta }{\int }}\frac{1}{t_1}\underset{0}{\overset{t_1}{\int }}\frac{1}{t_2}\underset{0}{\overset{t_2}{\int }}\dots \frac{1}{t_{n-1}}\underset{0}{\overset{t_{n-1}}{\int }}{t}_n^{b-1}f\left(t_n\right)d{t}_{n}d{t}_{n-1}\dots d{t}_1.$$

(19)

For $\zeta \in \mathbb{U},f\in A,n\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ and $b\in \mathbb{C}\setminus {\mathbb{Z}}^{-},$ Kutbi and Attiya [9] making use of $J_{s,b}\left(f\right)$ as a differential operator as

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

(20)

$$J_{-1,1}\left(f\right)\left(\zeta \right)=\frac{1}{2}\left(f\left(\zeta \right)+\zeta f^{{}^{\prime }}\left(\zeta \right)\right)$$

(21)

$$J_{-1,\frac{1}{1-\lambda }}\left(f\right)\left(\zeta \right)=\lambda f\left(\zeta \right)+(1-\lambda )\zeta f^{{}^{\prime }}\left(\zeta \right)(\lambda \ne 1)$$

(22)

$$J_{-n,0}\left(f\right)\left(\zeta \right)=D^n\left(f\right)\left(\zeta \right)$$

(23)

$$J_{-n,\frac{1}{\lambda }-1}\left(f\right)\left(\zeta \right)=D_{\lambda }^n\left(f\right)\left(\zeta \right)(\lambda \ne 0)$$

(24)

$$J_{-n,\lambda }\left(f\right)\left(\zeta \right)=I_{\lambda }^n\left(f\right)\left(\zeta \right)((\lambda >-1)$$

(25)

and

$$J_{-n,1}\left(f\right)\left(\zeta \right)=I_n\left(f\right)\left(\zeta \right)$$

(26)

More details for Srivastava–Attiya operator see for example ([7,9,10,11,12]).

Definition 1.

A function $f\left(\zeta \right)\in A$is said to be in the family $H(s,b,\varphi )$if it satisfies

$$(1+b)\frac{J_{s,b}\left(f\right)\left(\zeta \right)}{J_{s+1,b}\left(f\right)\left(\zeta \right)}-b\prec \varphi \left(\zeta \right)\left(\zeta \in \mathbb{U}\right)$$

where $\varphi \left(\zeta \right)$ is defined by (8), $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$ and $s\in \mathbb{C}.$

Let f be an analytic function which is a single-valued function in some domain $\mathbb{D}$ in complex plan $\mathbb{C},$ we say that f is univalent function, if it does not take the same value twice in $\mathbb{D}$; that is, if $f\left({\zeta }_1\right)\ne$$f\left({\zeta }_2\right)$ for a different values ${\zeta }_1$ and ${\zeta }_2$ in $\mathbb{D}$ (see ([3], page 26)). Additionally, a function $f\in A$ is said to be bi-univalent in $\mathbb{U}$if both functions f and $f^{-1}$ are univalent in $\mathbb{U}$.

Let $\mathsf{\Sigma }$ denote the class of bi-univalent functions in $\mathbb{U}$ defined by (1).

The class $\mathsf{\Sigma }$ was introduced and studied by Lewin [13] and he showed that $|a_2|<1.51$. Several authors have found non-sharp estimates of the coefficients $\left|a_2\right|$ and $\left|a_3\right|$ of Taylor–Maclaurin’s series. For examples of functions belong to $\mathsf{\Sigma },$see ([12,14]):

$$\frac{\zeta }{1-\zeta },-log(1-\zeta )\mathrm{and}\frac{1}{2}log\left(\frac{1+\zeta }{1-\zeta }\right).$$

For various subclasses of bi-univalent functions, see, e.g., ([15,16,17,18,19,20,21,22,23,24,25,26]).

Definition 2.

A function $f\in$Σ given by (1) is said to be in the family $H_{\mathsf{\Sigma }}(s,b,\varphi )$if both f and $f^{-1}$ are in $H(s,b,\varphi ).$

For examples of functions belong to $H_{\mathsf{\Sigma }}(s,b,\varphi )$:

1.

We can see the function $f\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }}\in \mathsf{\Sigma }$, then it is easy to see that $f\left(\zeta \right)\in H(-1,b,\frac{1}{1-\zeta })$, and we have, $g=$$f^{-1}\left(\zeta \right)={\displaystyle \frac{\zeta }{1+\zeta }}\in \mathsf{\Sigma }$, then

$$(1+b)\frac{J_{-1,b}\left(g\right)\left(\zeta \right)}{J_{0,b}\left(g\right)\left(\zeta \right)}-b=\frac{1}{1+\zeta },$$

the right hand side of the above equation is subordinate to $\varphi \left(\zeta \right)={\displaystyle \frac{1}{1-\zeta }}$. Therefore, $f\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }}\in H_{\mathsf{\Sigma }}\left(-1,b,\frac{1}{1-\zeta }\right).$

Additionally, we can see,

2.

$f\left(\zeta \right)=-log(1-\zeta )\in H_{\mathsf{\Sigma }}\left(-2,0,\frac{1+\zeta }{1-\zeta }\right).$

In our paper, we use the Faber polynomial expansion to estimate bounds of the coefficients $|a_n|$ of bi-univalent functions in the family $H_{\mathsf{\Sigma }}(s,b,\varphi ),$ also, we obtain the bounds of the initial coefficients of the functions in this family.

Unless otherwise mentioned, we assume throughout this paper that $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-},{\mathbb{Z}}_0^{-}={\mathbb{Z}}^{-}\cup \left\{0\right\}=\{0,-1,-2,\dots \},s\in \mathbb{C}when\zeta \in \mathbb{U}.$

2. Coefficient Estimates of ${\mathit{H}}_{\mathsf{\Sigma }}\mathbf{(}\mathit{s}\mathbf{,}\mathit{b}\mathbf{,}\mathbf{\varphi }\mathbf{)}$

Theorem 1.

Let the function $f\in \mathsf{\Sigma }$ given by (1) be in the family $H_{\mathsf{\Sigma }}(s,b,\varphi ).$ Additionally, let $a_m=0$ and $a_p\ne 0$ for $2\le m,p\le n,$where $(p-1)$ is a divisor of $(n-1).$ Then

$$\left|a_p\right|\le {\left(\frac{B_1}{p-1}\right)}^{\left(\frac{p-1}{n-1}\right)}exp\left(\left(1+Res\right)ln\left|\frac{1+b}{p+b}\right|-Ims\left(arg\left(1+b\right)-arg\left(p+b\right)\right)\right),p\ge 3,$$

where $B_1$ is defined in (8).

Proof. 

If we set $F\left(\zeta \right)=J_{s+1,b}\left(f\right)\left(\zeta \right),$ then

$F\left(\zeta \right)=\zeta +{\displaystyle \sum _{n=2}^{\infty }}{\delta }_n{\zeta }^n,$ with ${\delta }_n={\left(\frac{1+b}{n+b}\right)}^{s+1}{a}_n.$ Using relation (13), we have

$$f\in H_{\mathsf{\Sigma }}(s,b,\varphi )\mathit{if}\mathit{,}\mathit{and}\mathit{only}\mathit{if}\mathit{,}\frac{\zeta F^{{}^{\prime }}\left(\zeta \right)}{F\left(\zeta \right)}\prec \varphi \left(\zeta \right).$$

Since, f and its inverse $g=f^{-1}$ are in $H_{\mathsf{\Sigma }}(s,b,\varphi ),g\left(\zeta \right)=\zeta +{\displaystyle \sum _{n=2}^{\infty }}{b}_n{\zeta }^n,$ therefore, there are analytic functions u and $v$with $u\left(0\right)=v\left(0\right)=0,\left|u\right(\zeta \left)\right|<1$ and $\left|v\right(\zeta \left)\right|<1,$ such that

$$\frac{\zeta F^{{}^{\prime }}\left(\zeta \right)}{F\left(\zeta \right)}=\varphi \left(u\left(\zeta \right)\right)(\zeta \in \mathbb{U})$$

(27)

and

$$\frac{w{G}^{{}^{\prime }}\left(w\right)}{G\left(w\right)}=\varphi \left(v\left(w\right)\right)(\zeta \in \mathbb{U}),$$

(28)

where $G\left(\zeta \right)=J_{s+1,b}\left(g\right)\left(\zeta \right),$ then $G\left(\zeta \right)=\zeta +{\displaystyle \sum _{n=2}^{\infty }}{\zeta }_n{\zeta }^n$, with ${\zeta }_n={\left(\frac{1+b}{n+b}\right)}^{s+1}{b}_n,$ using (5) we have $b_n=\frac{1}{n}{K}_{n-1}^{-n}(a_2,a_3,\dots ,a_n).$

If the functions $u\left(\zeta \right)$ and $v\left(\zeta \right)$ defined by

$$u\left(\zeta \right)={\displaystyle \sum _{n=1}^{\infty }}{c}_n{\zeta }^{n}andv\left(\zeta \right)={\displaystyle \sum _{n=1}^{\infty }}{d}_n{\zeta }^n(\zeta \in \mathbb{U}).$$

(29)

It is well known that (see Duren ([3], page 265))

$$\left|c_n\right|\le 1\mathrm{and}\left|d_n\right|\le 1n=2,3,\dots .$$

(30)

Additionally, we have

$$\varphi \left(u\left(\zeta \right)\right)=1-B_1{\displaystyle \sum _{n=1}^{\infty }}{K}_n^{-1}(c_1,-c_2,\dots ,{(-1)}^{n+1}{c}_n,B_1,B_2,\dots ,B_n){\zeta }^n(\zeta \in \mathbb{U}),$$

(31)

and

$$\varphi \left(v\left(\omega \right)\right)=1-B_1{\displaystyle \sum _{n=1}^{\infty }}{K}_n^{-1}(d_1,-d_2,\dots ,{(-1)}^{n+1}{d}_n,B_1,B_2,\dots ,B_n)w^n(\zeta \in \mathbb{U}).$$

(32)

In general (see ([20], page 649)), the coefficients $K_n^p:={\mathcal{K}}_n^p(k_1,k_2,\dots ,k_n,B_1,B_2,B_3,\dots ,B_n)$ are given by

$$\begin{array}{cc}\hfill & {\mathcal{K}}_n^p=\frac{p!}{(p-n)!n!}{k}_1^n\frac{B_n}{B_1}+\frac{p!}{(p-n+1)!(n-2)!}{k}_1^{n-2}{k}_2\frac{B_{n-1}}{B_1}\hfill \\ \hfill & +\frac{p!}{(p-n+2)!(n-3)!}{k}_1^{n-3}{k}_3\frac{B_{n-2}}{B_1}\hfill \\ \hfill & +\frac{p!}{(p-n+3)!(n-4)!}{k}_1^{n-4}[k_4\frac{B_{n-3}}{B_1}+\frac{p-n+3}{2}{k}_2^2{k}_3\frac{B_{n-2}}{B_1}]\hfill \\ \hfill & +{\displaystyle \sum _{j\ge 5}^{\infty }}{k}_1^{n-j}{X}_j,\hfill \end{array}$$

where $X_j$ is a homogeneous polynomial of degree j in the variables $k_2,\dots ,k_n.$

Using the Faber polynomial expansion (2) yield the following identities

$$\frac{\zeta F^{{}^{\prime }}\left(\zeta \right)}{F\left(\zeta \right)}=1-{\displaystyle \sum _{n=2}^{\infty }}{F}_{n-1}({\delta }_2,{\delta }_3,\dots ,{\delta }_n){\zeta }^{n-1},$$

(33)

and

$$\frac{w{G}^{{}^{\prime }}\left(w\right)}{G\left(w\right)}=1-{\displaystyle \sum _{n=2}^{\infty }}{F}_{n-1}({\zeta }_2,{\zeta }_3,\dots ,{\zeta }_n)w^{n-1}.$$

(34)

Comparing the corresponding coefficients of (33) and (31) yields

$$F_{n-1}({\delta }_2,{\delta }_3,\dots ,{\delta }_n)=B_1{K}_{n-1}^{-1}(c_1,-c_2,\dots ,{(-1)}^n{c}_{n-1},B_1,B_2,B_3,\dots ,B_{n-1})$$

(35)

and similarly, from (34) and (32), we have

$$F_{n-1}({\zeta }_2,{\zeta }_3,\dots ,{\zeta }_n)=B_1{K}_{n-1}^{-1}(d_1,-d_2,\dots ,{(-1)}^n{d}_{n-1},B_1,B_2,B_3,\dots ,B_{n-1}).$$

(36)

Since $a_m=0$ and $a_p\ne 0$ for $2\le m,p\le n,$and $(p-1)$ is a divisor of $(n-1),$ then by using (35) and (36), we have,

$$(p-1){(-1)}^{\left(\frac{n-1}{p-1}\right)}{\left({\delta }_p\right)}^{\left(\frac{n-1}{p-1}\right)}=-B_1{(-1)}^p{c}_{p-1}$$

(37)

and

$$(p-1){(-1)}^{\left(\frac{n-1}{p-1}\right)}{\left({\zeta }_p\right)}^{\left(\frac{n-1}{p-1}\right)}=-B_1{(-1)}^p{d}_{p-1},$$

(38)

also, under the condition $a_m=0$ and $a_p\ne 0$ for $2\le m,p\le n$ and using the relation between ${\delta }_p$ and ${\zeta }_p$ we have ${\zeta }_p=-{\delta }_p.$ Then (37) get

$$(p-1){(-1)}_p^{\left(\frac{n-1}{p-1}\right)}{\left(-{\delta }_p\right)}^{\left(\frac{n-1}{p-1}\right)}=-B_1{(-1)}^p{d}_{p-1},$$

(39)

Using either (37) or (39), we have $\left|{\delta }_p\right|\le {\left(\frac{B_1}{p-1}\right)}^{\left(\frac{p-1}{n-1}\right)},$ substitute of ${\delta }_p={\left(\frac{1+b}{p+b}\right)}^{s+1}{a}_p$ and with some simple calculation we get the theorem. □

Noting that the result in Theorem 1 is sharp for function $f\left(\zeta \right)$ given by

$$f\left(\zeta \right)=\zeta +{\left(\frac{B_1}{p-1}\right)}^{\left(\frac{p-1}{n-1}\right)}exp\left(\left(1+Res\right)ln\left|\frac{1+b}{p+b}\right|-Ims\left(arg\left(1+b\right)-arg\left(p+b\right)\right)\right){\zeta }^n,$$

(40)

where $n\ge 3$ and $B_1$ is defined in (8).

Putting $p=n$ in Theorem 1, we have

Corollary 1.

Let the function $f\in \mathsf{\Sigma }$ given by (1) be in the class $S_{\mathsf{\Sigma }}(\alpha ,\beta ,\gamma ,k,\varphi ),$ if $a_m=0$ for $2\le m\le n-1.$ Then

$$\left|a_n\right|\le \left(\frac{B_1}{n-1}\right)exp\left(\left(1+Res\right)ln\left|\frac{1+b}{n+b}\right|-Ims\left(arg\left(1+b\right)-arg\left(n+b\right)\right)\right),n\ge 3,$$

(41)

where $B_1$ is defined in (8). The result is sharp for function $f\left(\zeta \right)$ given by (40).

To prove our next theorem, we shall need the following lemma.

Lemma 2.

([3,20],) Let the function $\mathsf{\Phi }\left(\zeta \right)={\displaystyle \sum _{n=1}^{\infty }}{\mathsf{\Phi }}_n{\zeta }^n$ be a Schwarz function with $\left|\mathsf{\Phi }\left(\zeta \right)\right|<1$, $\zeta \in \mathbb{U}$. Then for $-\infty <\eta <\infty ,$

$$\left|{\mathsf{\Phi }}_2+\eta {\mathsf{\Phi }}_1^2\right|\le \left\{\begin{array}{c}1-(1-\eta )\left|{\mathsf{\Phi }}_1^2\right|\eta >0\\ \\ 1-(1+\eta )\left|{\mathsf{\Phi }}_1^2\right|\eta \le 0\end{array}\right.$$

Theorem 2.

Let the function $f\in \mathsf{\Sigma }$ given by (1) be in the class $S_{\mathsf{\Sigma }}(\alpha ,\beta ,\gamma ,k,\varphi ).$ Then

$$\left|a_2\right|\le \left\{\begin{array}{c}2\sqrt{B_1}exp\left(\left(1+Res\right)ln\left|\frac{2+b}{1+b}\right|-Ims\left(arg\left(2+b\right)-arg\left(1+b\right)\right)\right)\left((B_1\ge \left|B_2\right|)\right)\\ \\ 2\sqrt{B_2}exp\left(\left(1+Res\right)ln\left|\frac{2+b}{1+b}\right|-Ims\left(arg\left(2+b\right)-arg\left(1+b\right)\right)\right)(B_1<\left|B_2\right|)\end{array}\right.$$

(42)

and

$$\left|a_3-a_2^2\right|\le \left\{\begin{array}{c}B1\left(\left|\frac{3}{4\mathcal{M}}-\frac{1}{2{\mathcal{L}}^2}\right|+\left|\frac{1}{4\mathcal{M}}-\frac{1}{2{\mathcal{L}}^2}\right|\right)(B_1\ge \left|B_2\right|)\\ \\ \left|B2\right|\left(\left|\frac{3}{4\mathcal{M}}-\frac{1}{2{\mathcal{L}}^2}\right|+\left|\frac{1}{4\mathcal{M}}-\frac{1}{2{\mathcal{L}}^2}\right|\right)(B_1<\left|B_2\right|),\end{array}\right.$$

(43)

where

$$\mathcal{L}=\frac{1}{2}exp\left(\left(1+Res\right)ln\left|\frac{1+b}{2+b}\right|-Ims\left(arg\left(1+b\right)-arg\left(2+b\right)\right)\right)$$

and

$$\mathcal{M}=\frac{1}{6}exp\left(\left(1+Res\right)ln\left|\frac{1+b}{3+b}\right|-Ims\left(arg\left(1+b\right)-arg\left(3+b\right)\right)\right).$$

Proof. 

Putting $n=2$ and $n=3$ in (35) and (36), respectively, we find that

$${\delta }_2=c_1{B}_1,$$

(44)

$$-{\zeta }_2=d_1{B}_1,$$

$${\delta }_2^2-2{\delta }_3=c_1^2{B}_2+c_2{B}_1,$$

(45)

and

$${\zeta }_2^2-2{\zeta }_3=d_1^2{B}_2+d_2{B}_1.$$

(46)

Moreover, we have ${\zeta }_2=-{\delta }_2$ and ${\zeta }_3=2{\delta }_2^2-{\delta }_2.$

Therefore, (45) and (46) imply

$$a_2^2=\frac{-B_1}{2{\mathcal{L}}^2}\left(\left(c_2+\frac{B_2}{B_1}{c}_1^2\right)+\left(d_2+\frac{B_2}{B_1}{d}_1^2\right)\right)$$

(47)

Applying Lemma 2.

Case I. If $B_2>0.$ Then both $\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|$ and $\left|d_2+\frac{B_2}{B_1}{d}_1^2\right|$ have maximum value at $1$ when $B_1>B_2$. Additionally,

$\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|$ and $\left|d_2+\frac{B_2}{B_1}{d}_1^2\right|$ have maximum value at $\frac{B_2}{B_1}$ when $B_1\le B_2,$ where $\left|c_1\right|\le 1$ and $\left|d_1\right|\le 1.$

Case II. If $B_2\le 0,$ Then both $\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|$ and $\left|d_2+\frac{B_2}{B_1}{d}_1^2\right|$ have maximum value at $1$ when $B_1>-B_2$ and

$\left|c_2+\frac{B_2}{B_1}{c}_1^2\right|$ and $\left|d_2+\frac{B_2}{B_1}{d}_1^2\right|$ have maximum value at $\frac{-B_2}{B_1}$ when $B_1\le -B_2.$ Then, we have (42).

Moreover, from (45) and (46) we get

$$a_3=-\left(\frac{3c_2+d_2}{4\mathcal{M}}\right)B_1-\left(\frac{3c_1^2+d_1^2}{4\mathcal{M}}\right)B_2.$$

(48)

It follows from (47) and (48) that

$$a_3-a_2^2=-\left(\frac{3}{4\mathcal{M}}-\frac{1}{2{\mathcal{L}}^2}\right)\left(c_2+\frac{B_2}{B_1}{c}_1^2\right)B_1-\left(\frac{1}{4\mathcal{M}}-\frac{1}{2{\mathcal{L}}^2}\right)\left(d_2+\frac{B_2}{B_1}{d}_1^2\right)B_1.$$

(49)

Applying the cases mentioned above for $B_2>0$ and $B_2\le 0,$ we have (43). This completes the proof of the theorem. □

3. Conclusions

In this paper, we investigated a new family $H_{\mathsf{\Sigma }}(s,b,\varphi )$ of normalized analytic functions and bi-univalent functions associated with Srivastava–Attiya operator in the unit disk $\mathbb{U}.$ For the functions in the family $H_{\mathsf{\Sigma }}(s,b,\varphi ),$ we obtained bounds for the general coefficients $|a_n|$ of functions in $H_{\mathsf{\Sigma }}(s,b,\varphi )$. Moreover, we estimate initial Taylor–Maclaurin’s coefficient inequalities for the functions in $H_{\mathsf{\Sigma }}(s,b,\varphi )$.