The Study of the New Classes of m-Fold Symmetric bi-Univalent Functions

Abstract

In this paper, we introduce three new subclasses of m-fold symmetric holomorphic functions in the open unit disk $U,$ where the functions f and $f^{-1}$ are m-fold symmetric holomorphic functions in the open unit disk. We denote these classes of functions by $F{S}_{\Sigma ,m}^{p,q,s}(d)$, $F{S}_{\Sigma ,m}^{p,q,s}(e)$ and $F{S}_{\Sigma ,m}^{p,q,s,h,r}.$ As the Fekete-Szegö problem for different classes of functions is a topic of great interest, we study the Fekete-Szegö functional and we obtain estimates on coefficients for the new function classes.

1. Introduction and Preliminary Results

Let $\mathcal{A}$ denote the family of functions of the form

$$f(z)=z+\sum _{k=2}^{\infty }{a}_k{z}^k$$

(1)

which are analytic in the open unit disk $U=\{z\in \mathbb{C}:|z|<1\}$ and normalized by the conditions $f(0)=0,f^{\prime }(0)=1.$

Let $S\subset \mathcal{A}$ denote the subclass of all functions in $\mathcal{A}$ which are univalent in U (see [1]).

In [1], the Koebe one-quarter theorem ensures that the image of the unit disk under every $f\in S$ function contains a disk of radius $1/4$.

It is well known that every function $f\in S$ has an inverse $f^{-1}$, which is defined by

$$f^{-1}(f(z))=z,z\in U$$

and

$$f(f^{-1}(w))=w,|w|<r_0(f),r_0(f)\ge 1/4,$$

where

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

(2)

A function $f\in \mathcal{A}$ is said to be bi-univalent in U if both f and $f^{-1}$ are univalent in U. Let $\Sigma$ denote the class of all bi-univalent functions in U given by (1).

The class of bi-univalent functions was first introduced and studied by Lewin [2] and it was shown that $|a_2|<1.51.$

The domain D is m-fold symmetric if a rotation of D about the origin through an angle $2\pi /m$ carries D on itself.

We said that the holomorphic function f in the domain D is m-fold symmetric if the following condition is true: $f(e^{\frac{2\pi i}{m}}z)=e^{\frac{2\pi i}{m}}f(z).$

A function is said to be m-fold symmetric if it has the following normalized form:

$$f(z)=z+\sum _{k=1}^{\infty }{a}_{mk+1}{z}^{mk+1},z\in U,m\in \mathbb{N}\cup \{0\}.$$

(3)

The normalized form of f is given as in (3) and the series expansion for $f^{-1}(z)$ is given below (see [3]):

$$\begin{array}{c}g(w)=f^{-1}(w)=w-a_{m+1}{w}^{m+1}\\ +[(m+1)a_{m-1}^2-a_{2m+1}]w^{2m+1}\\ -[\frac{1}{2}(m+1)(3m+2)a_{m+1}^3-(3m+2)a_{m+1}{a}_{2m+1}+a_{3m+1}]w^{3m+1}+\dots \end{array}$$

(4)

We can give examples of m-fold symmetric bi-univalent functions: ${\{\frac{z^m}{1-z^m}\}}^{\frac{1}{m}};$ ${[-log(1-z^m)]}^{\frac{1}{m}};\frac{1}{2}log{(\frac{1+z^m}{1-z^m})}^{\frac{1}{m}}.$

The important results about the m-fold symmetric analytic bi-univalent functions are given in [3,4,5,6,7].

The Fekete-Szegö problem is the problem of maximizing the absolute value of the functional $|a_3-\mu a_2^2|.$

Fekete-Szegö inequalities for different classes of functions are studied in the papers [8,9,10,11,12,13,14].

Many authors obtained coefficient estimates of bi-univalent functions in the articles [2,14,15,16,17,18,19,20,21,22,23,24,25].

Definition 1.

Let $f\in \mathcal{A}$ be given by (1) and $0<q<p\le 1.$ Then, the $(p,q)$-derivative operator for the function f of the form (1) is defined by

$$D_{p,q}f(z)=\frac{f(pz)-f(qz)}{(p-q)z},z\in U^{\ast }=U-\{0\}$$

(5)

and

$$(D_{p,q}f)(0)=f^{\prime }(0)$$

(6)

and it follows that the function f is differentiable at 0.

We deduce from (2) that

$$D_{p,q}f(z)=1+\sum _{k=2}^{\infty }{[k]}_{p,q}{a}_k{z}^{k-1}$$

(7)

where the $(p,q)$-bracket number is given by

$${[k]}_{p,q}=\frac{p^k-q^k}{p-q}=p^{k-1}+p^{k-2}q+p^{k-3}{q}^2+\dots +p{q}^{k-2}+q^{k-1},p\ne q$$

which is a natural generalization of the q-number.

Too ${lim}_{p\to 1^{-}}{[k]}_{p,q}={[k]}_q=\frac{1-q^k}{1-q}$, see [26,27].

Definition 2

([28]). Let the function $f\in \mathcal{A},where0\le d<1,s\ge 1$ is real. The function $f\in L_s(d)$ of s-pseudo-starlike function of order d in the unit disk U if and only if

$$Re(\frac{z{[f^{\prime }(z)]}^s}{f(z)})>d.$$

Lemma 1

([1], p. 41). Let the function $w\in \mathcal{P}$ be given by the following series: $w(z)=1+w_{1}z+w_2{z}^2+\dots ,z\in U$, where we denote by $\mathcal{P}$ the class of Carathéodory functions analytic in the open disk U,

$$\mathcal{P}=\{w\in \mathcal{A}|w(0)=1,Re(w(z))>0,z\in U\}.$$

The sharp estimate given by $|w_n|\le 2,n\in {\mathbb{N}}^{\ast }$ holds true.

2. Main Results

Definition 3.

The function f given by (3) is in the function class $F{S}_{\Sigma ,m}^{p,q,s}(d)(m\in \mathbb{N},$ $0<q<p\le 1,s\ge 1,0<d\le 1,(z,w)\in U)$ if:

$$\left\{\begin{array}{c}f\in \Sigma ,\hfill \\ |arg{(D_{p,q}f(z))}^s|<\frac{d\pi }{2},z\in U\hfill \end{array}\right.$$

(8)

and

$$|arg{(D_{p,q}g(w))}^s|<\frac{d\pi }{2},w\in U,$$

(9)

where g is the function given by (4).

Remark 1.

In the case when $m=1$ (one-fold case) and $s=1$, we obtain the class defined in [29].

Remark 2.

In the case when $p=1,$ we obtain $li{m}_{q\to 1^{-}}F{S}_{\Sigma ,1}^1(d)=F{S}_{\Sigma }(d)$, the class which was introduced by Srivastava et al. in [24].

We obtain coefficient bounds for the functions class $F{S}_{\Sigma ,m}^{p,q,s}(d)$ in the next theorem.

Theorem 1.

Let f given by (3) be in the class $F{S}_{\Sigma ,m}^{p,q,s}(d)(m\in \mathbb{N},0<q<p\le 1,s\ge 1,$ $0<d\le 1,(z,w)\in U).$ Then,

$$|a_{m+1}|\le \frac{2d}{\sqrt{sd(m+1){[2m+1]}_{p,q}-s(d-s){[m+1]}_{p,q}^2}}$$

(10)

and

$$|a_{2m+1}|\le \frac{2d}{s{[2m+1]}_{p,q}}+\frac{2(m+1)d^2}{s^2{[m+1]}_{p,q}^2}.$$

(11)

Proof. 

If we use the relations (8) and (9), we obtain

$${(D_{p,q}f(z))}^s={[\alpha (z)]}^d$$

(12)

and

$${(D_{p,q}g(w))}^s={[\beta (w)]}^d,(z,w\in U)$$

(13)

where the functions $\alpha (z)$ and $\beta (w)$ are in $\mathcal{P}$ and are given by

$$\alpha (z)=1+{\alpha }_m{z}^m+{\alpha }_{2m}{z}^{2m}+{\alpha }_{3m}{z}^{3m}+\dots$$

(14)

and

$$\beta (w)=1+{\beta }_m{w}^m+{\beta }_{2m}{w}^{2m}+{\beta }_{3m}{w}^{3m}+\dots .$$

(15)

It is obvious that

$${[\alpha (z)]}^d=1+d{\alpha }_m{z}^m+(d{\alpha }_{2m}+\frac{d(d-1)}{2}{\alpha }_m^2)z^{2m}+\dots ,$$

$${[\beta (w)]}^d=1+d{\beta }_m{w}^m+(d{\beta }_{2m}+\frac{d(d-1)}{2}{\beta }_m^2)w^{2m}+\dots ,$$

$${(D_{p,q}f(z))}^s=1+s{[m+1]}_{p,q}{a}_{m+1}{z}^m$$

$$+(s{[2m+1]}_{p,q}{a}_{2m+1}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2)z^{2m}+\dots$$

and

$$\begin{array}{c}{(D_{p,q}g(w))}^s=1-s{[m+1]}_{p,q}{a}_{m+1}{w}^m\\ -s{[2m+1]}_{p,q}{a}_{2m+1}{w}^{2m}+(s(m+1){[2m+1]}_{p,q}{a}_{m+1}^2+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2)w^{2m}+\dots \end{array}$$

If we compare the coefficients in the relations (12) and (13), we have

$$s{[m+1]}_{p,q}{a}_{m+1}=d{\alpha }_m,$$

(16)

$$s{[2m+1]}_{p,q}{a}_{2m+1}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2$$

$$=d{\alpha }_{2m}+\frac{d(d-1)}{2}{\alpha }_m^2,$$

(17)

$$-s{[m+1]}_{p,q}{a}_{m+1}=d{\beta }_m,$$

(18)

$$-s{[2m+1]}_{p,q}{a}_{2m+1}+(s(m+1){[2m+1]}_{p,q}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2)a_{m+1}^2$$

$$=d{\beta }_{2m}+\frac{d(d-1)}{2}{\beta }_m^2.$$

(19)

We obtain from the relations (16) and (18)

$${\alpha }_m=-{\beta }_m$$

(20)

and

$$2s^2{[m+1]}_{p,q}^2{a}_{m+1}^2=d^2({\alpha }_m^2+{\beta }_m^2)$$

(21)

Now, from the relations (17), (19) and (21), we obtain that

$$s(s-1)d{[m+1]}_{p,q}^2{a}_{m+1}^2+(m+1)sd{[2m+1]}_{p,q}{a}_{m+1}^2$$

$$-(d-1)s^2{[m+1]}_{p,q}^2{a}_{m+1}^2=d^2({\alpha }_{2m}+{\beta }_{2m}).$$

We have

$$a_{m+1}^2=\frac{d^2({\alpha }_{2m}+{\beta }_{2m})}{s{[m+1]}_{p,q}^2(s-d)+(m+1)sd{[2m+1]}_{p,q}}.$$

(22)

If we apply Lemma 1 for the coefficients ${\alpha }_{2m}$ and ${\beta }_{2m}$, we have

$$|a_{m+1}|\le \frac{2d}{\sqrt{(m+1)sd{[2m+1]}_{p,q}-(d-s)s{[m+1]}_{p,q}^2}}.$$

If we use the relations (17) and (19), we obtain the next relation

$$\begin{array}{c}2s{[2m+1]}_{p,q}{a}_{2m+1}-s(m+1){[2m+1]}_{p,q}{a}_{m+1}^2\\ =d({\alpha }_{2m}-{\beta }_{2m})+\frac{d(d-1)}{2}({\alpha }_m^2-{\beta }_m^2).\end{array}$$

(23)

It follows from (20), (21) and (23) that

$$a_{2m+1}=\frac{(m+1)d^2({\alpha }_m^2+{\beta }_m^2)}{4s^2{[m+1]}_{p,q}^2}+\frac{d({\alpha }_{2m}-{\beta }_{2m})}{2s{[2m+1]}_{p,q}}.$$

(24)

If we apply Lemma 1 for the coefficients ${\alpha }_m,{\alpha }_{2m},{\beta }_m,{\beta }_{2m}$, we obtain

$$|a_{2m+1}|\le \frac{2d}{{[2m+1]}_{p,q}s}+\frac{2d^2(m+1)}{s^2{[m+1]}_{p,q}^2}.$$

 □

Remark 3.

For one-fold case $m=1$ and $s=1$ in Theorem 1, we obtain the results obtained in [29].

Remark 4.

For a one-fold case and $p=1$, we have

$$li{m}_{q\to 1^{-}}F{S}_{\Sigma ,1}^{q,1}(d)=F{S}_{\Sigma }(d),$$

the results of Srivastava et al. [24].

Definition 4.

The function f given by (3) is in the class $F{S}_{\Sigma ,m}^{p,q,s}(e)(0\le e<1,0<q<p\le 1,s\ge 1,(z,w)\in U,m\in \mathbb{N})$ if the following conditions are satisfied:

$$\left\{\begin{array}{c}f\in \Sigma ,\hfill \\ \mathcal{R}\{{(D_{p,q}f(z))}^s\}>e,z\in U\hfill \end{array}\right.$$

(25)

$$\mathcal{R}\{{(D_{p,q}g(w))}^s\}>e,w\in U,$$

(26)

where the function g is defined by Relation (4).

Remark 5.

For $m=1$ (one-fold case) and $s=1,$ we obtain the class of functions obtained in [29].

Remark 6.

When $p=1$, we obtain $li{m}_{q\to 1^{-}}F{S}_{\Sigma ,1}^1(e)=F{S}_{\Sigma }(d),$ the class which was introduced by Srivastava et al. in [24].

In the next theorem, we obtain coefficient bounds for the function class $F{S}_{\Sigma ,m}^{p,q,s}(e).$

Theorem 2.

Let the function f given by (3) be in the function class $F{S}_{\Sigma ,m}^{p,q,s}(e),(m\in \mathbb{N},$ $0<q<p\le 1,s\ge 1,0\le e<1,(z,w)\in U)$. Then,

$$|a_{m+1}|\le min\{\frac{2(1-e)}{s{[m+1]}_{p,q}},2\sqrt{\frac{(1-e)}{s(s-1){[m+1]}_{p,q}^2+(m+1)s{[2m+1]}_{p,q}}}\}$$

(27)

$$|a_{2m+1}|\le \frac{2(1-e)(m+1)}{s(s-1){[m+1]}_{p,q}^2+(m+1)s{[2m+1]}_{p,q}}+\frac{2(1-e)}{s{[2m+1]}_{p,q}}.$$

(28)

Proof. 

If we use Relations (25) and (26), we obtain

$${(D_{p,q}f(z))}^s=e+(1-e)\alpha (z)$$

(29)

and

$${(D_{p,q}g(w))}^s=e+(1-e)\beta (w),z,w\in U,$$

(30)

respectively, where

$$\alpha (z)=1+{\alpha }_m{z}^m+{\alpha }_{2m}{z}^{2m}+{\alpha }_{3m}{z}^{3m}+\dots$$

and

$$\beta (w)=1+{\beta }_m{w}^m+{\beta }_{2m}{w}^{2m}+{\beta }_{3m}{w}^{3m}+\dots ,$$

$\alpha (z)$ and $\beta (w)$ are in $\mathcal{P}.$

It is obvious that

$$e+(1-e)\alpha (z)=1+(1-e){\alpha }_m{z}^m+(1-e){\alpha }_{2m}{z}^{2m}+\dots ,$$

and

$$e+(1-e)\beta (w)=1+(1-e){\beta }_m{w}^m+(1-e){\beta }_{2m}{w}^{2m}+\dots$$

Already,

$${(D_{p,q}f(z))}^s=1+s{[m+1]}_{p,q}{a}_{m+1}{z}^m+(s{[2m+1]}_{p,q}{a}_{2m+1}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2)z^{2m}+\dots$$

and

$${(D_{p,q}g(w))}^s=1-s{[m+1]}_{p,q}{a}_{m+1}{w}^m-s{[2m+1]}_{p,q}{a}_{2m+1}{w}^{2m}$$

$$+(s(m+1){[2m+1]}_{p,q}{a}_{m+1}^2+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2)w^{2m}+\dots$$

From the relations (29) and (30), if we compare the coefficients, we obtain the following relations:

$$s{[m+1]}_{p,q}{a}_{m+1}=(1-e){\alpha }_m,$$

(31)

$$s{[2m+1]}_{p,q}{a}_{2m+1}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2=(1-e){\alpha }_{2m},$$

(32)

$$-s{[m+1]}_{p,q}{a}_{m+1}=(1-e){\beta }_m,$$

(33)

$$-s{[1+2m]}_{p,q}{a}_{2m+1}{+(s[2m+1]}_{p,q}(m+1)$$

$$+\frac{s(s-1)}{2}{[1+m]}_{p,q}^2)a_{m+1}^2=(1-e){\beta }_{2m}.$$

(34)

We obtain from Relations (31) and (33)

$${\alpha }_m=-{\beta }_m$$

(35)

and

$$2s^2{[m+1]}_{p,q}^2{a}_{m+1}^2={(1-e)}^2({\alpha }_m^2+{\beta }_m^2).$$

(36)

We obtain now from Relations (32) and (34) the following relation:

$$\begin{array}{c}s(s-1){[m+1]}_{p,q}^2{a}_{m+1}^2+(m+1)s{[2m+1]}_{p,q}{a}_{m+1}^2=\\ (1-e)({\alpha }_{2m}+{\beta }_{2m}).\end{array}$$

(37)

From Lemma 1 for the coefficients ${\alpha }_m,{\alpha }_{2m},{\beta }_m,{\beta }_{2m}$, we obtain that

$$|a_{m+1}|\le 2\sqrt{\frac{1-e}{(m+1)s{[2m+1]}_{p,q}+s(s-1){[m+1]}_{p,q}^2}}.$$

If we use Relations (32) and (34) to find the bound on $|a_{2m+1}|$, we obtain the following relation:

$$-s(1+m){[1+2m]}_{p,q}{a}_{m+1}^2+2s{[1+2m]}_{p,q}{a}_{2m+1}=(1-e)({\alpha }_{2m}-{\beta }_{2m}),$$

(38)

or equivalently

$$a_{2m+1}=\frac{(1-e)({\alpha }_{2m}-{\beta }_{2m})}{2s{[2m+1]}_{p,q}}+\frac{(m+1)}{2}{a}_{m+1}^2.$$

(39)

From Relation (36), if we substitute the value of $a_{m+1}^2$, we obtain

$$a_{2m+1}=\frac{(1-e)({\alpha }_{2m}-{\beta }_{2m})}{2s{[2m+1]}_{p,q}}+\frac{(m+1){(1-e)}^2({\alpha }_m^2+{\beta }_m^2)}{4s^2{[m+1]}_{p,q}^2}.$$

(40)

Now, if we apply Lemma 1 for the coefficients ${\alpha }_m,{\alpha }_{2m},{\beta }_m,{\beta }_{2m}$, we obtain

$$|a_{2m+1}|\le \frac{2(1-e)}{s{[2m+1]}_{p,q}}+\frac{2(m+1){(1-e)}^2}{s^2{[m+1]}_{p,q}^2}.$$

From Relations (37) and (39) applying Lemma 1, we obtain

$$|a_{2m+1}|\le \frac{2(m+1)(1-e)}{s(s-1){[m+1]}_{p,q}^2+(m+1)s{[2m+1]}_{p,q}}+\frac{2(1-e)}{s{[2m+1]}_{p,q}}.$$

 □

Remark 7.

For one fold case ($m=1$) and $s=1$ in Theorem 2, we obtain the results given in [29].

Remark 8.

For a one-fold case, in Theorem 2, choosing $p=1,q\to 1^{-}$, we obtain the following corollary.

Corollary 1.

[24] Let the function $f\in F{S}_{\Sigma }(e),(s=1,0\le e<1,(z,w)\in U)$ be given by (1). Then,

$$|a_2|\le \sqrt{\frac{2(1-e)}{3}}$$

and

$$|a_3|\le \frac{(1-e)(5-3e)}{3}.$$

In the following theorems, we provide the Fekete-Szegö type inequalities for the functions of the families $F{S}_{\Sigma ,m}^{p,q,s}(d)$ and $F{S}_{\Sigma ,m}^{p,q,s}(e).$

Theorem 3.

Let f be a function of the form (3) in the class $F{S}_{\Sigma ,m}^{p,q,s}(d)$. Then,

$$|a_{2m+1}-\sigma a_{m+1}^2|\le \left\{\begin{array}{c}\frac{2d}{s{[2m+1]}_{p,q}},|t(\sigma )|\le \frac{1}{s{[2m+1]}_{p,q}}\hfill \\ 4sd|t(\sigma )|,|t(\sigma )|\ge \frac{1}{s{[2m+1]}_{p,q}},\hfill \end{array}\right.$$

(41)

where

$$t(\sigma )=\frac{d(m+1-2\sigma )}{2s{[m+1]}_{p,q}^2(s-d)+2s(m+1)d{[2m+1]}_{p,q}}.$$

Proof. 

We want to calculate $a_{2m+1}-\sigma a_{m+1}^2.$

For this, from Relations (22) and (24), where we know the values of the coefficients $a_{m+1}^2$ and $a_{2m+1}:$

$$a_{m+1}^2=\frac{d^2({\alpha }_{2m}+{\beta }_{2m})}{s{[m+1]}_{p,q}^2(s-d)+(m+1)sd{[2m+1]}_{p,q}},$$

$$a_{2m+1}=\frac{(m+1)d^2({\alpha }_m^2+{\beta }_m^2)}{4s^2{[m+1]}_{p,q}^2}+\frac{d({\alpha }_{2m}-{\beta }_{2m})}{2s{[2m+1]}_{p,q}},$$

it follows that

$$a_{2m+1}-\sigma a_{m+1}^2=$$

$$d[{\alpha }_{2m}(\frac{1}{2s{[2m+1]}_{p,q}}+\frac{d(m+1-2\sigma )}{2s{[m+1]}_{p,q}^2(s-d)+2s(m+1)d{[2m+1]}_{p,q}})$$

$$+{\beta }_{2m}(\frac{d(m+1-2\sigma )}{2s{[m+1]}_{p,q}^2(s-d)+2sd(m+1){[2m+1]}_{p,q}}-\frac{1}{2s{[2m+1]}_{p,q}})].$$

According to Lemma 1 and after some computations, we obtain

$$|a_{2m+1}-\sigma a_{m+1}^2|\le \left\{\begin{array}{c}\frac{2d}{s{[2m+1]}_{p,q}},|t(\sigma )|\le \frac{1}{s{[2m+1]}_{p,q}}\hfill \\ 4sd|t(\sigma )|,|t(\sigma )|\ge \frac{1}{s{[2m+1]}_{p,q}}.\hfill \end{array}\right.$$

 □

Theorem 4.

Let f be a function of the form $(3)$ in the class $F{S}_{\Sigma ,m}^{p,q,s}(e)$. Then,

$$|a_{2m+1}-\sigma a_{m+1}^2|\le \left\{\begin{array}{c}\frac{2(1-e)}{s{[2m+1]}_{p,q}},|t(\sigma )|\le \frac{1}{2s{[2m+1]}_{p,q}}\hfill \\ 4s(1-e)|t(\sigma )|,|t(\sigma )|\ge \frac{1}{2s{[2m+1]}_{p,q}}\hfill \end{array}\right.,$$

(42)

where

$$t(\sigma )=\frac{(1-2\sigma +m)}{2s(s-1)d{[m+1]}_{p,q}^2+2s(m+1){[2m+1]}_{p,q}}.$$

Proof. 

We will compute $a_{2m+1}-\sigma a_{m+1}^2,$ using the values of the coefficients $a_{m+1}^2$ and $a_{2m+1}$ given in Relations (37) and (39).

It follows that

$$a_{2m+1}-\sigma a_{m+1}^2$$

$$=(1-e)[{\alpha }_{2m}(\frac{1}{2s{[2m+1]}_{p,q}}+\frac{1-2\sigma +m}{2s(s-1)d{[m+1]}_{p,q}^2+2(m+1)s{[2m+1]}_{p,q}})$$

$$+{\beta }_{2m}(\frac{(1+m-2\sigma )}{2s(s-1)d{[m+1]}_{p,q}^2+2s(m+1){[2m+1]}_{p,q}}-\frac{1}{2s{[2m+1]}_{p,q}})].$$

According to Lemma 1 and after some computations, we obtain

$$|a_{2m+1}-\sigma a_{m+1}^2|\le \left\{\begin{array}{c}\frac{2(1-e)}{s{[2m+1]}_{p,q}},|t(\sigma )|\le \frac{1}{2s{[2m+1]}_{p,q}}\hfill \\ 4s(1-e)|t(\sigma )|,|t(\sigma )|\ge \frac{1}{2s{[2m+1]}_{p,q}}\hfill \end{array}\right..$$

 □

Definition 5.

Let $h,r:U\to \mathbb{C}$ be analytic functions and $min\{Re(h(z)),Re(r(z))\}>0,$ where $z\in U,h(0)=r(0)=1.$

A function f given by $(3)$ is said to be in the class $F{S}_{\Sigma ,m}^{p,q,s,h,r}$, where $s\ge 1,0<q<p\le 1,m\in \mathbb{N}$ if the conditions are satisfied:

$${\left(D_{p,q}f(z)\right)}^s\in h(U),z\in U$$

(43)

and

$${\left(D_{p,q}g(w)\right)}^s\in r(U),w\in U,$$

(44)

where the function g is given by $(4)$.

We obtain coefficient bounds for the functions class $F{S}_{\Sigma ,m}^{p,q,s,h,r}$ in the following theorem.

Theorem 5.

Let the function f given by $(3)$ be in the class $F{S}_{\Sigma ,m}^{p,q,s,h,r}$. Then,

$$|a_{m+1}|\le min\{\sqrt{\frac{|h_1^{\prime }{(0)|}^2+{|r_1^{\prime }(0)|}^2}{2s^2{[m+1]}_{p,q}^2}},\sqrt{\frac{|h_2^{″}(0)|+|r_2^{″}(0)|}{s(s-1){[m+1]}_{p,q}^2+s(m+1){[2m+1]}_{p,q}}}\};$$

(45)

$$|a_{2m+1}|\le min\{\frac{(|h^{\prime }(0){|}^2+|r^{\prime }(0){|}^2)(m+1)}{4s^2{[m+1]}_{p,q}^2}+\frac{|h^{″}(0)|+|t^{″}(0)|}{2s{[2m+1]}_{p,q}},$$

$$\frac{|h^{″}(0)|+|r^{″}(0)|}{2s{[2m+1]}_{p,q}}+\frac{(m+1)(|h^{″}(0)|+|r^{″}(0)|)}{2s\{(m+1){[2m+1]}_{p,q}+(s-1){[m+1]}_{p,q}^2\}}\}.$$

(46)

Proof. 

In Relations $(43)$ and $(44)$, the equivalent forms of the argument inequalities are

$${(D_{p,q}f(z))}^s=h(z),$$

(47)

and

$${(D_{p,q}g(w))}^s=r(w),$$

(48)

where $h(z)$ and $r(w)$ satisfy the conditions from Definition 5, and have the following Taylor–Maclaurin series expansions:

$$h(z)=1+h_{1}z+h_2{z}^2+\dots$$

(49)

$$r(w)=1+r_{1}w+r_2{w}^2+\dots$$

(50)

If we substitute (49) and (50) into (47) and (48), respectively, and equate the coefficients, we obtain

$$s{[m+1]}_{p,q}{a}_{m+1}=h_1;$$

(51)

$$s{[2m+1]}_{p,q}{a}_{2m+1}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2{a}_{m+1}^2=h_2;$$

(52)

$$-s{[m+1]}_{p,q}{a}_{m+1}=r_1;$$

(53)

$$-s{[2m+1]}_{p,q}{a}_{2m+1}+(s(m+1){[2m+1]}_{p,q}+\frac{s(s-1)}{2}{[m+1]}_{p,q}^2)a_{m+1}^2=r_2.$$

(54)

We obtain that

$$h_1=-r_1$$

(55)

and

$$h_1^2+r_1^2=2s^2{[m+1]}_{p,q}^2{a}_{m+1}^2$$

(56)

from Relations (51) and (53).

Adding Relations (52) and (54), we obtain that

$$a_{m+1}^2\{s(s-1){[m+1]}_{p,q}^2+s(m+1){[2m+1]}_{p,q}\}=h_2+r_2.$$

(57)

Now, from (56) and (57), we obtain

$$a_{m+1}^2=\frac{h_1^2+r_1^2}{2s^2{[m+1]}_{p,q}^2}$$

(58)

$$a_{m+1}^2=\frac{h_2+r_2}{s(s-1){[m+1]}_{p,q}^2+s(m+1){[2m+1]}_{p,q}}.$$

(59)

We obtain from Relations (58) and (59) that

$$|a_{m+1}{|}^2\le \frac{|h_1^{\prime }{(0)|}^2+{|r_1^{\prime }(0)|}^2}{2s^2{[m+1]}_{p,q}^2}$$

and

$$|a_{m+1}{|}^2\le \frac{|h_2^{″}(0)|+|r_2^{″}(0)|}{s(s-1){[m+1]}_{p,q}^2+s(m+1){[2m+1]}_{p,q}}.$$

So, we obtain the estimate on the coefficient $|a_{m+1}|$ as in (45).

Next, substracting (54) from (52), we obtain the following relation:

$$2s{[2m+1]}_{p,q}{a}_{2m+1}-s(m+1){[2m+1]}_{p,q}{a}_{m+1}^2=h_2-r_2.$$

(60)

Substituting the value of $a_{m+1}^2$ from (58) into (60), it follows that

$$a_{2m+1}=\frac{h_2-r_2}{2s{[2m+1]}_{p,q}}+\frac{(m+1)(h_1^2+r_1^2)}{4s^2{[m+1]}_{p,q}^2}.$$

Therefore,

$$|a_{2m+1}|\le \frac{(|h^{\prime }(0){|}^2+|r^{\prime }(0){|}^2)(m+1)}{4s^2{[m+1]}_{p,q}^2}+\frac{|h^{″}(0)|+|t^{″}(0)|}{2s{[2m+1]}_{p,q}}.$$

Upon substituting the value of $a_{m+1}^2$ from (59) into (60), it follows that

$$a_{2m+1}=\frac{h_2-r_2}{2s{[2m+1]}_{p,q}}+\frac{(m+1)(h_2+r_2)}{\{(s-1){[m+1]}_{p,q}^2+(m+1){[2m+1]}_{p,q}\}2s}.$$

So, it follows that

$$|a_{2m+1}|\le \frac{|h^{″}(0)|+|r^{″}(0)|}{2s{[2m+1]}_{p,q}}+\frac{(m+1)(|h^{″}(0)|+|r^{″}(0)|)}{2s\{(m+1){[2m+1]}_{p,q}+(s-1){[m+1]}_{p,q}^2\}}.$$

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3. Conclusions

As future research directions, the symmetry properties of this operator, the $(p,q)$-derivative operator, can be studied.