The study of coefficient estimates and Fekete–Szegö inequalities for the new classes of m-fold symmetric bi-univalent functions defined using an operator

Abstract

The objective of this paper is to introduce new classes of m-fold symmetric bi-univalent functions. We discuss estimates on the Taylor–Maclaurin coefficients $|a_{m+1}|$ and $|a_{2m+1}|$, and the Fekete–Szegő problem is also considered for the new classes of functions introduced. We denote these classes by $MF-S_{\Sigma ,m}^{p,q}(h)$, $MF-S_{\Sigma , m}^{p,q}(s)$, and $MF-S_{\Sigma , m}^{b,d}$. Quantum calculus aspects are also considered in this study to enhance its novelty and to obtain more interesting results.

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 $f(0)=0$, $f'(0)=1$.

The subclass $S\subset \mathcal{A}$ is formed of all functions in the class $\mathcal {A}$ that are univalent in U (see[14]).

The Koebe one-quarter theorem ensures that the image of the unit disk under every $f\in S$ function contains a disk of radius $1/4$, see [14].

If the function $f\in S$, then it has an inverse $f^{-1}$, which is defined by

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

and

$$ f\bigl(f^{-1}(w)\bigr)=w, \quad \vert w \vert < r_{0}(f), r_{0}(f)\geq{1/4}, $$

where

$$ g(w)=f^{-1}(w)=w-a_{2}w^{2}+ \bigl(2a_{2}^{2}-a_{3}\bigr)w^{3}- \bigl(5a_{2}^{3}-5a_{2}a_{3}+a_{4} \bigr)w^{4}+\cdots . $$

(2)

We say that a function $f\in \mathcal {A}$ is bi-univalent in U if both f and $f^{-1}$ are univalent in U.

We denote by Σ the class of all bi-univalent functions in U given by (1).

The study on bi-univalent functions has its origins in the article published by Lewin in [25], where 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.

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

Definition 1

([36])

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

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

(3)

The normalized form of f is given as in (3), and the series expansion for $f^{-1}(z)$ is given below (see[4]):

$$\begin{aligned} g(w)={}&f^{-1}(w)=w-a_{m+1}w^{m+1} \\ &{}+\bigl[(m+1)a_{m-1}^{2}-a_{2m+1} \bigr]w^{2m+1} \\ &{}-\biggl[\frac{1}{2}(m+1) (3m+2)a_{m+1}^{3}-(3m+2)a_{m+1}a_{2m+1}+a_{3m+1} \biggr]w^{3m+1}+\cdots . \end{aligned}$$

(4)

Examples of m-fold symmetric bi-univalent functions are:

$$\bigl[-\log\bigl(1-z^{m}\bigr)\bigr]^{\frac{1}{m}}; \qquad \biggl\{ \frac{z^{m}}{1-z^{m}} \biggr\} ^{\frac{1}{m}}; \qquad { \frac{1}{2}\log \biggl(\frac{1+z^{m}}{1-z^{m}}\biggr)^{ \frac{1}{m}}}. $$

Srivastava et al. in the paper [36] defined m-fold symmetric bi-univalent functions following the concept of m-fold symmetric univalent functions.

The interest in bi-univalent functions resurfaced in 2010 when a paper authored by Srivastava et al. in [35] was published. It opened the door for many interesting developments on the topic. Soon other new subclasses of bi-univalent functions were introduced [19–21] and special classes of bi-univalent functions were investigated such as Ma–Minda starlike and convex functions [3], analytic bi-Bazilevic functions [23], and recently a family of bi-univalent functions associated with Bazilevic functions and the λ-pseudo-starlike functions [38]. Brannan and Clunie’s conjecture [8] was further investigated [32] and subordination properties were also obtained for certain subclasses of bi-univalent functions [11]. New results continued to emerge in the recent years such as coefficient estimates for some general subclasses of analytic and bi-univalent functions [13, 27, 34]. Horadam polynomials were used for applications on Bazilevic bi-univalent functions satisfying subordination conditions [40] and for introducing certain classes of bi-univalent functions [1]. Operators were also included in the study as it can be seen in earlier publications [9] and in very recent ones [28]. Interesting results regarding m-fold symmetric bi-univalent functions were published in the same year when this notion was introduced [21]. This continued to appear in the following years [4, 16, 31, 33] and is still researched today [10, 37], proving that the topic remains in development.

The Fekete–Szegö problem is the problem of maximizing the absolute value of the functional $|a_{3}-\mu a_{2}^{2}|$.

The Fekete–Szegö inequalities introduced in 1933, see [18], preoccupied researchers regarding different classes of univalent functions [15, 24]. Hence it is obvious that such inequalities were obtained regarding bi-univalent functions too and very recently published papers can be cited to support the assertion that the topic still provides interesting results [2, 6, 41]. Inspiring new results emerged when quantum calculus was involved in the studies, as can be seen in many papers [30] and in studies published very recently [5, 12, 17, 39]. Some elements of the $(p,q)$-calculus must be used for obtaining the original results contained in this paper. Further information can be found in [22, 30]. The tremendous impact quantum calculus has had when associated with univalent functions theory is nicely highlighted in the recent review paper [22, 30].

For obtaining the original results contained in this paper, some elements of the $(p,q)$-calculus must be used.

Definition 2

([22], p. 2)

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

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

(5)

and

$$ (D_{p,q}f) (0)=f'(0), $$

(6)

it follows that the function f is differentiable at 0.

We can deduce from relation (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

$$ [k]_{p,q}=\frac{p^{k}-q^{k}}{p-q}=p^{k-1}+p^{k-2}q+p^{k-3}q^{2}+ \cdots+pq^{k-2}+q^{k-1},\quad p \neq q $$

which is a natural generalization of the q-number.

We can see that $\lim_{p\rightarrow 1^{-}}[k]_{p,q}=[k]_{q}=\frac{1-q^{k}}{1-q}$, see the papers [17, 22].

Definition 3

([7], p.137)

Let the function $f\in \mathcal{A}$, where $0\leq d<1$, $s\geq 1$ is real. The function $f\in L_{s}(d)$ of an s-pseudo-starlike function of order d in the unit disk U if and only if

$$ \operatorname{Re}\biggl(\frac{z[f'(z)]^{s}}{f(z)}\biggr)>d. $$

Lemma 4

[14, 29] Let the function $w\in \mathcal {P}$ be given by the following series $w(z)=1+w_{1}z+w_{2} z^{2}+\cdots$ , $z\in U$, where we denote by $\mathcal{P}$ the class of Carathéodory functions analytic in the open disk U,

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

The sharp estimate given by $|w_{n}|\leq 2$, $n\in \mathbb{N}^{*}$ holds true.

In the next section of the paper, the original results obtained are presented in three definitions of new subclasses of m-fold symmetric bi-univalent functions and theorems concerning coefficient estimates and Fekete–Szegő problem for the newly defined classes.

2 Main results

Definition 5

The function class $M-FS_{\Sigma ,m}^{p,q}(h), (m\in \mathbb{N}, 0< q< p\leq 1, 0< h\leq 1, (z,w) \in U)$, contains all the functions f given by relation (3) that satisfy the following conditions:

$$ \textstyle\begin{cases} f\in \Sigma _{m} \\ \vert \arg\{D_{p,q}f(z)+z(D_{p,q}f(z))^{\prime} \} \vert < \frac{h \pi}{2},& (z\in U)\end{cases} $$

(8)

and

$$ \bigl\vert \arg\bigl\{ D_{p,q}g(w)+w\bigl(D_{p,q}g(w) \bigr)^{\prime} \bigr\} \bigr\vert < \frac{h \pi}{2}, $$

(9)

where g is given by relation (4).

The coefficient bounds for the functions class $MF-S_{\Sigma ,m}^{p,q}(h)$ are obtained in the next theorem.

Theorem 6

If the function f, given by relation (3), is in the function class $MF-S_{\Sigma ,m}^{p,q}(h), (m\in \mathbb{N}, 0< q< p\leq 1, 0< h\leq 1, (z,w) \in U)$, then the following inequalities are true:

$$ \vert a_{m+1} \vert \leq \frac{2h}{\sqrt{(m+1)h[2m+1]_{p,q}(1+2m)-(h-1)(1+m)^{2}[m+1]^{2}_{p,q}}} $$

(10)

and

$$ \vert a_{2m+1} \vert \leq \frac{2h}{(1+2m)[2m+1]_{p,q}}+ \frac{2h^{2}}{(1+m)[m+1]^{2}_{p,q}}. $$

(11)

Proof

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

$$ D_{p,q}f(z)+z\bigl(D_{p,q}f(z)\bigr)^{\prime} = \bigl[\alpha (z)\bigr]^{h},\quad z\in U $$

(12)

and

$$ D_{p,q}g(w)+w\bigl(D_{p,q}g(w) \bigr)^{\prime} =\bigl[\beta (w)\bigr]^{h},\quad w\in U, $$

(13)

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

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

(14)

and

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

(15)

Comparing the coefficients in (12) and (13), we obtain

$$\begin{aligned}& (1+m)[m+1]_{p,q}a_{m+1}=h\alpha _{m}, \end{aligned}$$

(16)

$$\begin{aligned}& (1+2m)[2m+1]_{p,q}a_{2m+1} =h\alpha _{2m}+\frac{h(h-1)}{2}\alpha _{m}^{2}, \end{aligned}$$

(17)

$$\begin{aligned}& -(1+m)[m+1]_{p,q}a_{m+1}=h\beta _{m}, \end{aligned}$$

(18)

$$\begin{aligned}& (1+2m)[2m+1]_{p,q}\bigl((m+1)a^{2}_{m+1}-a_{2m+1} \bigr) =h\beta _{2m}+\frac{h(h-1)}{2}\beta _{m}^{2}. \end{aligned}$$

(19)

From (16) and (18) we obtain

$$ \alpha _{m}=-\beta _{m} $$

(20)

and

$$ 2(1+m)^{2}[m+1]_{p,q}^{2} a_{m+1}^{2}=h^{2}\bigl(\alpha _{m}^{2}+\beta _{m}^{2} \bigr). $$

(21)

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

$$\begin{aligned}& (m+1) (1+2m)[2m+1]_{p,q}a_{m+1}^{2}\\& \quad =h(\alpha _{2m}+\beta _{2m})+(h-1)\biggl[ \frac{(1+m)^{2}[m+1]^{2}_{p,q}}{h}\biggr] a_{m+1}^{2}. \end{aligned}$$

Therefore, we obtain that

$$\begin{aligned}& a_{m+1}^{2} = \frac{h^{2}(\alpha _{2m}+\beta _{2m})}{(m+1)(1+2m)[2m+1]_{p,q}h-(h-1)(1+m)^{2}[m+1]^{2}_{p,q}}. \end{aligned}$$

Now, for the coefficients $\alpha _{2m}$ and $\beta _{2m}$, if we apply Lemma 4, we obtain

$$ \vert a_{m+1} \vert \leq \frac{2h}{\sqrt{h(m+1)(1+2m)[2m+1]_{p,q}-(h-1)(1+m)^{2}[m+1]^{2}_{p,q}}}. $$

If we use (17) and (19), then we obtain

$$\begin{aligned}& 2(1+2m)[2m+1]_{p,q}a_{2m+1}-(m+1) (1+2m)[2m+1]_{p,q}a^{2}_{m+1} \\& \quad =h(\alpha _{2m}-\beta _{2m})+\frac{h(h-1)}{2}\bigl( \alpha _{m}^{2}-\beta _{m}^{2} \bigr). \end{aligned}$$

(22)

From (20), (21), and (22), we obtain

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

(23)

If we apply Lemma 4 for the coefficients $\alpha _{m}$, $\alpha _{2m}$, $\beta _{m}$, $\beta _{2m}$, we obtain

$$ \vert a_{2m+1} \vert \leq \frac{2h}{(1+2m)[2m+1]_{p,q}}+ \frac{2h^{2}}{(1+m)[m+1]^{2}_{p,q}}. $$

 □

The Fekete–Szegö functional for the class $MF-S_{\Sigma , m}^{p,q}(h)$ is given in the next theorem.

Theorem 7

Let f be a function of the form (3) in the class $MF-S_{\Sigma , m}^{p,q}(h)$. Then

$$ \bigl\vert a_{2m+1}-\rho a_{m+1}^{2} \bigr\vert \leq \textstyle\begin{cases} \frac{2h}{(1+2m)[2m+1]_{p,q}},& \vert l(\rho ) \vert \leq \frac{1}{(1+2m)[2m+1]_{p,q}}, \\ 4h(1+2m)[2m+1]_{p,q}^{2} \vert l(\rho ) \vert , & \vert l(\rho ) \vert \geq \frac{1}{(1+2m)[2m+1]_{p,q}}, \end{cases} $$

(24)

where we denote

$$ l(\rho )= \frac{h\{m+1-2\rho \}}{2\{h[2m+1]_{p,q}(1+2m)-[m+1]^{2}_{p,q}(h-1)(1+m)\}}. $$

Proof

The values of the coefficients $a_{m+1}^{2}$ and $a_{2m+1}$ are given in the proof of Theorem 6 as follows:

$$\begin{aligned}& a_{2m+1}=\frac{h(\alpha _{2m}-\beta _{2m})}{2(1+2m)[2m+1]_{p,q}}+ \frac{h^{2}(\alpha _{m}^{2}+\beta _{m}^{2})}{4(1+m)[m+1]^{2}_{p,q}}, \\& a_{m+1}^{2}= \frac{h^{2}(\alpha _{2m}+\beta _{2m})}{h(m+1)(1+2m)[2m+1]_{p,q}-(h-1)(1+m)^{2}[m+1]^{2}_{p,q}}. \end{aligned}$$

We start to compute $a_{2m+1}-\rho a_{m+1}^{2}$.

It follows that

$$\begin{aligned}& a_{2m+1}-\rho a_{m+1}^{2} \\& \quad =h\{\alpha _{2m}\biggl[\frac{1}{2(1+2m)[2m+1]_{p,q}} \\& \qquad {}+ \frac{h(m+1-2\rho )}{ 2\{[2m+1]_{p,q}(1+2m)h-(h-1)(1+m)[m+1]^{2}_{p,q}\}}\biggr]\\& \qquad {}+\beta _{2m}\biggl[-\frac{1}{2(1+2m)[2m+1]_{p,q}}\\& \qquad {}+ \frac{h(m+1-2\rho )}{2[h(1+2m)[2m+1]_{p,q}-(h-1)(1+m)[m+1]^{2}_{p,q}]} \biggr]. \end{aligned}$$

After some computations and according to Lemma 4, we obtain

$$ \bigl\vert a_{2m+1}-\rho a_{m+1}^{2} \bigr\vert \leq \textstyle\begin{cases} \frac{2h}{(1+2m)[2m+1]_{p,q}}, & \vert l(\rho ) \vert \leq \frac{1}{(1+2m)[2m+1]_{p,q}}, \\ 4h(1+2m)[2m+1]_{p,q}^{2} \vert l(\rho ) \vert , & \vert l(\rho ) \vert \geq \frac{1}{(1+2m)[2m+1]_{p,q}}. \end{cases} $$

 □

Definition 8

The function class $MF-S_{\Sigma , m}^{p,q}(s), (0< q< p\leq 1, 0\leq s<1, m\in \mathbb{N}, (z, w)\in U)$, contains all the functions f given by relation (3) that satisfy the following conditions:

$$\begin{aligned}& \textstyle\begin{cases} f\in \Sigma _{m} \\ \operatorname{Re}\{D_{p,q}f(z)+z(D_{p,q}f(z))^{\prime} \}>s, \quad z\in U \end{cases}\displaystyle \end{aligned}$$

(25)

$$\begin{aligned}& \operatorname{Re}\bigl\{ D_{p,q}g(w)+w\bigl(D_{p,q}g(w) \bigr)^{\prime} \bigr\} >s, \quad w\in U \end{aligned}$$

(26)

where the function g is of the form (4).

Coefficient bounds for the functions class $MF-S_{\Sigma , m}^{p,q}(s)$ are obtained in the next theorem.

Theorem 9

Let f be a function in the class $MF-S_{\Sigma , m}^{p,q}(s), (m\in \mathbb{N}, 0< q< p\leq 1, 0\leq s<1, (z, w)\in U)$, which has the form (3). Then

$$ \vert a_{m+1} \vert \leq \min \biggl\{ \frac{2(1-s)^{2}}{(1+m)^{2}[m+1]^{2}_{p,q}},2 \sqrt{\frac{(1-s)}{(m+1)(1+2m)[2m+1]_{p,q}}}\biggr\} $$

(27)

and

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

(28)

Proof

We can see that from (24) and (25) we obtain

$$ D_{p,q}f(z)+z\bigl(D_{p,q}f(z)\bigr)^{\prime} =s+(1-s)\alpha (z),\quad z\in U $$

(29)

and

$$ D_{p,q}g(w)+w\bigl(D_{p,q}g(w)\bigr)^{\prime} =s+(1-s)\beta (w),\quad w\in U, $$

(30)

where $\alpha (z)$ and $\beta (w)$ in $\mathcal {P}$ are given by (14) and (15).

Now we compare the coefficients from (28) and (29), and we obtain

$$\begin{aligned}& (1+m)[m+1]_{p,q}a_{m+1}=(1-s)\alpha _{m}, \end{aligned}$$

(31)

$$\begin{aligned}& (1+2m)[2m+1]_{p,q}a_{2m+1}=(1-s)\alpha _{2m}, \end{aligned}$$

(32)

$$\begin{aligned}& -(1+m)[m+1]_{p,q}a_{m+1}=(1-s)\beta _{m}, \end{aligned}$$

(33)

$$\begin{aligned}& (1+2m)[2m+1]_{p,q}\bigl[(m+1)a_{m+1}^{2}-a_{2m+1} \bigr] =(1-s)\beta _{2m}. \end{aligned}$$

(34)

We obtain from (30) and (32) that

$$ \alpha _{m}=-\beta _{m} $$

(35)

and

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

(36)

From (33) and (31) we obtain

$$\begin{aligned}& (1+2m)[2m+1]_{p,q}(m+1)a_{m+1}^{2} =(1-s) (\alpha _{2m}+\beta _{2m}). \end{aligned}$$

(37)

If we apply Lemma 4 for the coefficients $\alpha _{m}$, $\alpha _{2m}$, $\beta _{m}$, $\beta _{2m}$, then we obtain

$$ \vert a_{m+1} \vert \leq 2\sqrt{\frac{1-s}{[2m+1]_{p,q}(m+1)(1+2m)}}. $$

Using (33) and (31) to find the bound on $|a_{2m+1}|$, we obtain

$$ \begin{gathered}[b] -(m+1) (1+2m)[2m+1]_{p,q}a_{m+1}^{2}+2(1+2m)[2m+1]_{p,q}a_{2m+1}\\ \quad =(1-s) ( \alpha _{2m}-\beta _{2m}), \end{gathered} $$

(38)

or equivalently

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

(39)

From (35) we substitute the value of $a_{m+1}^{2}$ and obtain

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

(40)

Now, we will apply Lemma 4 for the coefficients $\alpha _{m}$, $\alpha _{2m}$, $\beta _{m}$, $\beta _{2m}$, and we obtain

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

From (36) and (38), if we apply again Lemma 4, then we obtain

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

 □

In the next theorem we compute the Fekete–Szegö functional for the class $MF-S_{\Sigma , m}^{p,q}(s)$.

Theorem 10

Let f be a function of the form (3) in the class $MF-S_{\Sigma , m}^{p,q}(s)$. Then

$$ \bigl\vert a_{2m+1}-\rho a_{m+1}^{2} \bigr\vert \leq \textstyle\begin{cases} \frac{2(1-s)}{(1+2m)[2m+1]_{p,q}}, & \vert l(\rho ) \vert \leq \frac{1}{2(1+2m)[2m+1]_{p,q}}, \\ 4(1+2m)(1-s)[2m+1]_{p,q}^{2} \vert l(\rho ) \vert , & \vert l(\rho ) \vert \geq \frac{1}{2(1+2m)[2m+1]_{p,q}}, \end{cases} $$

(41)

where $l(\rho )$ is given by

$$ l(\rho )= \frac{(1-s)(1+2m)[2m+1]_{p,q}-4\rho [m+1]^{2}_{p,q}}{4(1+m)[m+1]^{2}_{p,q}(1+2m)[2m+1]_{p,q}}. $$

Proof

Using the values of $a_{m+1}^{2}$ and $a_{2m+1}$ from the proof of Theorem 9, we can compute $a_{2m+1}-\rho a_{m+1}^{2}$.

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

We obtain

$$\begin{aligned}& a_{2m+1}-\rho a_{m+1}^{2} \\& \quad =(1-s) \biggl\{ \alpha _{2m} \biggl[\frac{1}{2[1+2m]_{p,q}(1+2m)} \\& \qquad {}+ \frac{(1-s)(2m+1)[1+2m]_{p,q}-4\rho [m+1]^{2}_{p,q}}{4(1+m)[m+1]_{p,q}^{2}(1+2m)[2m+1]_{p,q}} \biggr] \\& \qquad {}+\beta _{2m} \biggl[ \frac{(1-s)(1+2m)[2m+1]_{p,q}-4\rho [m+1]_{p,q}^{2}}{4(1+m)[m+1]^{2}_{p,q}(1+2m)[2m+1]_{p,q}} \\& \qquad {}-\frac{1}{2[1+2m]_{p,q}(1+2m)} \biggr] \biggr\} . \end{aligned}$$

The next inequality is obtained after some computations and according to Lemma 4:

$$ \bigl\vert a_{2m+1}-\rho a_{m+1}^{2} \bigr\vert \leq \textstyle\begin{cases} \frac{2(1-s)}{(1+2m)[2m+1]_{p,q}}, & \vert l(\rho ) \vert \leq \frac{1}{2(1+2m)[2m+1]_{p,q}}, \\ 4(1+2m)(1-s)[2m+1]_{p,q}^{2} \vert l(\rho ) \vert , &\vert l(\rho ) \vert \geq \frac{1}{2(1+2m)[2m+1]_{p,q}}. \end{cases} $$

 □

Definition 11

Let $b, d:U\rightarrow \mathbb{C}$ be analytic functions with the property $\min \{\operatorname{Re}(b(z)), \operatorname{Re}(d(z))\}>0$, where $z\in U$, $b(0)=d(0)=1$.

The class $MF-S_{\Sigma , m}^{b,d}$ contains all the functions f given by (3) if the following conditions are satisfied:

$$ \bigl(D_{p,q}f(z)+z\bigl(D_{p,q}f(z)\bigr)^{\prime} \bigr)\in b(U),\quad z\in U $$

(42)

and

$$ \bigl(D_{p,q}g(w)+w\bigl(D_{p,q}g(w)\bigr)^{\prime} \bigr)\in d(U),\quad w\in U $$

(43)

where the function g is given by (4).

In the next theorem we obtain the coefficient bounds for the function class $MF-S_{\Sigma , m}^{b,d}$.

Theorem 12

If the function f of the form (3) is in the class $MF-S_{\Sigma , m}^{b,d}$, then the following inequalities are satisfied:

$$\begin{aligned}& \vert a_{m+1} \vert \leq \min \biggl\{ \sqrt{ \frac{ \vert b_{1}'(0) \vert ^{2}+ \vert d_{1}'(0) \vert ^{2}}{2(1+m)^{2}[m+1]_{p,q}^{2}}}, \sqrt{ \frac{ \vert b_{2}^{\prime \prime} (0) \vert + \vert d_{2}^{\prime \prime} (0) \vert }{(1+2m)(m+1)[2m+1]_{p,q}}} \biggr\} \end{aligned}$$

(44)

$$\begin{aligned}& \textit{and} \end{aligned}$$

(45)

$$\begin{aligned}& \begin{aligned}[b] \vert a_{2m+1} \vert \leq{} &\min \biggl\{ \frac{( \vert b^{\prime} (0) \vert ^{2}+ \vert d^{\prime} (0) \vert ^{2})}{4(1+m)[2m+1]_{p,q}^{2}}+ \frac{ \vert b^{\prime \prime} (0) \vert ^{2}+ \vert d^{\prime \prime} (0) \vert ^{2}}{2(1+2m)[2m+1]_{p,q}},\\ &\frac{ \vert b^{\prime \prime} (0) \vert + \vert d^{\prime \prime} (0) \vert }{2(1+2m)[2m+1]_{p,q}}+ \frac{( \vert b^{\prime \prime} (0) \vert + \vert d^{\prime \prime} (0) \vert )}{ 2(1+2m)[2m+1]_{p,q}}\biggr\} . \end{aligned} \end{aligned}$$

(46)

Proof

We can write relations (42) and (43) as follows:

$$ D_{p,q}f(z)+z\bigl(D_{p,q}f(z)\bigr)^{\prime} =b(z) $$

(47)

and

$$ D_{p,q}g(w)+w\bigl(D_{p,q}g(w)\bigr)^{\prime} =d(w), $$

(48)

where the functions b and d have the following forms and satisfy the conditions from Definition 11:

$$\begin{aligned}& b(z)=1+b_{1} z+b_{2} z^{2}+\cdots, \end{aligned}$$

(49)

$$\begin{aligned}& d(w)=1+d_{1} w+d_{2} w^{2}+\cdots. \end{aligned}$$

(50)

Substituting relations (49) and (50) into (47) and (48), respectively, and equating the coefficients, we obtain

$$\begin{aligned}& (1+m)[m+1]_{p,q} a_{m+1}=b_{1}; \end{aligned}$$

(51)

$$\begin{aligned}& (1+2m)[2m+1]_{p,q} a_{2m+1}=b_{2}; \end{aligned}$$

(52)

$$\begin{aligned}& -(1+m)[m+1]_{p,q} a_{m+1}=d_{1}; \end{aligned}$$

(53)

$$\begin{aligned}& (1+2m)[2m+1]_{p,q}\bigl((m+1)a^{2}_{m+1}-a_{2m+1} \bigr)=d_{2}. \end{aligned}$$

(54)

We obtain from (51) and (53) that

$$ b_{1}=-d_{1} $$

(55)

and

$$ b_{1}^{2}+d_{1}^{2}=2(1+m)^{2}[m+1]_{p,q}^{2} a_{m+1}^{2}. $$

(56)

Adding relations (52) and (54), we obtain

$$ \bigl\{ (1+2m) (m+1)[2m+1]_{p,q}\bigr\} a_{m+1}^{2}=b_{2}+d_{2}. $$

(57)

From (56) and (57), we obtain

$$ a_{m+1}^{2}=\frac{b_{1}^{2}+d_{1}^{2}}{2(1+m)^{2}[m+1]^{2}_{p,q}} $$

(58)

and

$$ a_{m+1}^{2}=\frac{b_{2}+d_{2}}{(1+2m)(m+1)[2m+1]_{p,q}}. $$

(59)

We find from (58) and (59) that

$$ \vert a_{m+1} \vert ^{2}\leq \frac{ \vert b_{1}'(0) \vert ^{2}+ \vert d_{1}'(0) \vert ^{2}}{2(1+m)^{2}[m+1]_{p,q}^{2}} $$

and

$$ \vert a_{m+1} \vert ^{2}\leq \frac{ \vert b_{2}^{\prime \prime} (0) \vert + \vert d_{2}^{\prime \prime} (0) \vert }{(1+2m)(m+1)[2m+1]_{p,q}}. $$

We get in this way the desired estimate on the coefficient $|a_{m+1}|$ as asserted in (44).

By subtracting (54) from (52), we obtain

$$\begin{aligned}& 2(1+2m)[2m+1]_{p,q}a_{2m+1} -(1+2m)[2m+1]_{p,q}(m+1)a_{m+1}^{2} \\& \quad =b_{2}-d_{2}. \end{aligned}$$

(60)

It follows that

$$\begin{aligned}& a_{2m+1}= \frac{b_{2}-d_{2}}{2(1+2m)[2m+1]_{p,q}}+ \frac{b_{1}^{2}+d_{1}^{2}}{4(1+m)[2m+1]^{2}_{p,q}}, \end{aligned}$$

using the value of $a_{m+1}^{2}$ from (58) into (60).

Hence,

$$ \vert a_{2m+1} \vert \leq \frac{( \vert b^{\prime} (0) \vert ^{2}+ \vert d^{\prime} (0) \vert ^{2})}{4(1+m)[2m+1]_{p,q}^{2}}+ \frac{ \vert b^{\prime \prime} (0) \vert ^{2}+ \vert d^{\prime \prime} (0) \vert ^{2}}{2(1+2m)[2m+1]_{p,q}}. $$

Using in (60) $a_{m+1}^{2}$ given by (59), we have

$$\begin{aligned}& a_{2m+1}= \frac{b_{2}-d_{2}}{2(1+2m)[2m+1]_{p,q}}+ \frac{b_{2}+d_{2}}{2(1+2m)[2m+1]_{p,q}}. \end{aligned}$$

It follows that

$$\begin{aligned}& \vert a_{2m+1} \vert \leq \frac{ \vert b^{\prime \prime} (0) \vert + \vert d^{\prime \prime} (0) \vert }{2(1+2m)[2m+1]_{p,q}}+ \frac{ \vert b^{\prime \prime} (0) \vert + \vert d^{\prime \prime} (0) \vert )}{ 2(1+2m)[2m+1]_{p,q}}. \end{aligned}$$

 □

3 Conclusion

These classes of functions introduced in this paper can be extended and similar properties to those presented can be studied. Using the same research method as in the paper [36], we introduce in Definitions 5, 8, and 11 three new classes of m-fold symmetric bi-univalent functions. As future research, using other operators or the $(p,q)$-derivative operator, properties of starlikeness, convexity, and close-to-convexity of the new classes of functions could be investigated, and we can study the properties of symmetry of $(p,q)$-derivative operator. We believe that this study will motivate a number of researchers to extend this idea for other functions and classes of functions.