1. Introduction
Let \(\mathcal{A}\) be the family of holomorphic functions, normalized by the conditions \(f(0)=f'(0)-1=0\) which is of the form
\begin{equation}
\label{main}
f(z)=z+\rho_2z^2+\rho_3z^3+\cdots
\end{equation}
(1)
in the open unit disk \(\varOmega=\{z;z\in \mathbb{C}\;\text{and}\;|z|< 1\}\). We denote by \(\mathcal{G}\) the subclass of functions in \(\mathcal{A} \) which are univalent in \(\varOmega\) (for more details see [
1]).
The Keobe-One Quarter Theorem [1] state that the image of \(\varOmega\) under all univalent function \(f\in \mathcal{A}\) contains a disk of radius \(\frac{1}{4}\). Hence all univalent function \(f\in \mathcal{A}\) has an inverse \(f^{-1}\) satisfy \(f^{-1}(f(z))\) and \(f(f^{-1}(\upsilon))=\upsilon\) \((|\upsilon|< r_0(f),\;r_0(f)\ge\frac{1}{4})\), where
\begin{equation}
\label{main2}
g(\upsilon)=f^{-1}(\upsilon)=\upsilon-\rho_2\upsilon^2+(2\rho_2^2-\rho_3)\upsilon^3-(5\rho_2^3-5\rho_2\rho_3+\rho_4)\upsilon^4+\cdots
\end{equation}
(2)
A function \(f\in \mathcal{A}\) denoted by \(\varSigma\) is said to be bi-univalent in \(\varOmega\) if both \(f^{-1}(z)\) ans \(f(z)\) are univalent in \(\varOmega\) (see for details [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]).
A domain \(\varPsi\) is said to be \(m\)-fold symmetric if a rotation of \(\varPsi\) about the origin through an angle \(2\pi/m\) carries \(\varPsi\) on itself. Therefore, a function \(f(z)\) holomorphic in \(\varOmega\) is said to be \(m\)-fold symmetric if
\begin{equation*}
f\left(e^{\frac{2\pi i}{m}}z\right)
=e^{\frac{2\pi i}{m}}f(z).\end{equation*}
A function is said to be \(m\)-fold symmetric if it has the following normalized form
\begin{equation}
\label{4}
f(z)=z+\sum_{\phi=1}^{\infty}\rho_{m\phi+1}z^{m\phi+1}\qquad(z\in \varOmega,\;\; m\in\mathcal{N}=\{1,2,3,\cdots\}).
\end{equation}
(3)
Let \(\mathfrak{S}_m\) the class of \(m\)-fold symmetric univalent functions in \(\varOmega\), that are normalized by (3), in which, the functions in the class \(\mathfrak{S}\) are \(one\)-fold symmetric. Similar to the concept of \(m\)-fold symmetric univalent functions, we introduced the concept of \(m\)-fold symmetric bi-univalent functions which is denoted by \(\varSigma_m\). Each of the function \(f\in \varSigma\) produces \(m\)-fold symmetric bi-univalent function for each integer \(m\in\mathcal{N}\).
The normalized form of \(f(z)\) is given as in (3) and the series expansion for \(f^{-1}(z)\), which has been investigated by Srivastava et al., [12], is given below:
\begin{align}
\label{1.4}
g(\upsilon)=&f^{-1}(\upsilon)\notag\\=&\upsilon-\rho_{m+1}\upsilon^{m+1}+\left[(m+1)\rho^2_{m-1}-\rho_{2m+1}\right]\upsilon^{2m+1}\notag\\&-\Biggl[\frac{1}{2}(m+1)(3m+2)\rho^3_{m+1}-(3m+2)\rho_{m+1}\rho_{2m+1}+\rho_{3m+1}\Biggr].
\end{align}
(4)
Some of the examples of \(m\)-fold symmetric bi-univalent functions are
\[\Biggl\{\frac{z^m}{1-z^m}\Biggr\}^{\frac{1}{m}},\] \[\left[-\log(1-z^{m})\right]^{\frac{1}{m}},\] and \[\Biggl\{\frac{1}{2}\log \left(\frac{1+z^m}{1-z^m}\right)^{\frac{1}{m}}\Biggr\}.\]
For more details on \(m\)-fold symmetric analytic bi-univalent functions (see [
5,
12,
13,
14,
15,
16,
17]).
Jackson [18,19] introduced the \(q\)-derivative operator \(\mathcal{D}_q\) of a function as follows;
\begin{equation}
\label{a2}
\mathcal{D}_{q}f(z)=\frac{f(qz)-f(z)}{(q-1)z}
\end{equation}
(5)
and \(\mathcal{D}_qf(0)=f'(0)\). In case of \(g(z)=z^{k}\) for \(k\) is a positive integer, the \(q\)-derivative of \(f(z)\) is given by
\begin{equation*}
\mathcal{D}_{q}z^{k}=\frac{z^{k}-(zq)^{k}}{(q-1)z}=[k]_qz^{k-1}.
\end{equation*}
As \(q\longrightarrow1^{-}\) and \(k\in \mathcal{N}\), we get
\begin{equation}
\label{a3}
[k]_q=\frac{1-q^{k}}{1-q}=1+q+\cdots+q^{k}\longrightarrow k,
\end{equation}
(6)
where \((z\neq 0,\;q\neq0)\). For more details on the concepts of \(q\)-derivative (see [
5,
20,
21,
22,
23,
24,
25,
26,
27]).
Definition 1. [28]
Let \(f(z)\in \mathcal{A}\), \(0\leq\chi< 1\) and \(\sigma\ge 1\) is real. Then \(f(z)\in L_{\sigma}(\chi)\) of \(\sigma\)-pseodu-starlike function of order \(\chi\) in \(\varOmega\) if and only if
\begin{equation}
\Re\left(\frac{z[f'(z)]^{\sigma}}{f(z)}\right)>\chi.
\end{equation}
(7)
Babalola [
28] verified that, all pseodu-starlike function are Bazilevic of type \(\left(1-\frac{1}{\sigma}\right)\), order \(\chi^{\frac{1}{\sigma}}\) and univalent in \(\varOmega\).
Lemma 1. [1]
Let the function \(\omega\in \mathcal{P}\) be given by the following series
\(\omega(z)=1+\omega_1z+\omega_2z^2+\cdots\quad(z\in
\varOmega).\)
The sharp estimate given by
\(|\omega_n|\leq2\quad(n\in \mathcal{N})\)
holds true.
In [29] Girgaonkar et al., introduced a new subclasses of holomorphic and bi-univalent functions as follows:
Definition 2.
A function \(f(z)\) given by (1) is said to be in the class \(\mathcal{M}_{\varSigma}(\chi)\;(0< \chi\leq1,(z,\upsilon)\in\varOmega)\) if \(
f\in\mathcal{E}\), \(|\arg(f'(z))^{\sigma}|< \frac{\chi\pi}{2}
\)
and
\(
|\arg(g'(\upsilon))^{\sigma}|< \frac{\chi\pi}{2},
\)
where \(g(\upsilon)\) is given by (2).
Definition 3.
A function \(f(z)\) given by (1) is said to be in the class \(\mathcal{M}_{\varSigma}(\psi)\;(0\leq\psi\psi
\)
and
\( \Re[(g'(\upsilon))^{\sigma}]>\psi,\)
where \(g(\upsilon)\) is given by (2).
In this current research, we introduced two new subclasses denoted by \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\) and \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\) of the function class \(\varSigma_m\) and obtain estimates coefficient \(|\rho_{m+1}|\) and \(|\rho_{2m+1}|\) for functions in these two new subclasses.
2. Main 4esults
Definition 4.
A function \(f(z)\) given by (3) is said to be in the class \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\;(m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) if
\begin{equation}
\label{2.1}
f\in\varSigma\quad and \quad |\arg(\mathcal{D}_{q}f(z))^{\sigma}|< \frac{\chi\pi}{2},
\end{equation}
(8)
and
\begin{equation}
\label{2.2}
|\arg(\mathcal{D}_{q}g(\upsilon))^{\sigma}|< \frac{\chi\pi}{2},
\end{equation}
(9)
where \(g(\upsilon)\) is given by (2).
Remark 1.
We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{\sigma}_{\varSigma,1}(\chi)=\mathcal{M}^{\sigma}_{\varSigma}(\chi)\) which was introduced and studied by Girgaonkar et al., [29].
Remark 2.
We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{1}_{\varSigma,1}(\chi)=\mathcal{M}_{\varSigma}(\chi)\) which was introduced and studied by Srivastava et al., [11].
Theorem 1.
Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\), \((m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (3). Then
\begin{equation}
|\rho_{m+1}|\leq\frac{2\chi}{\sqrt{(m+1)\sigma\chi[2m+1]_q-(\chi-\sigma)\sigma[m+1]_q^2}},
\end{equation}
(10)
and
\begin{equation}
|\rho_{2m+1}|\leq\frac{2\chi}{\sigma[2m+1]_q}+\frac{2(m+1)\chi^2}{\sigma^2[m+1]_q^2}.
\end{equation}
(11)
Proof.
Using inequalities (1) and (9), we get
\begin{equation}
\label{2.5}
(\mathcal{D}_{q}f(z))^{\sigma}=[\tau(z)]^{\chi},
\end{equation}
(12)
and
\begin{equation}
\label{2.6}
(\mathcal{D}_{q}g(\upsilon))^{\sigma}=[\varsigma(\upsilon)]^{\chi}
\end{equation}
(13)
respectively, where \(\tau(z)\) and \(\varsigma(\upsilon)\) in \(\mathcal{P}\) are given by the following series
\begin{equation}
\label{2.6a}
\tau(z)=1+\tau_mz^m+\tau_{2m}z^{2m}+\tau_{3m}z^{3m}+\cdots,
\end{equation}
(14)
and
\begin{equation}
\label{2.7a}
\varsigma(\upsilon)=1+\varsigma_m\upsilon^m+\varsigma_{2m}\upsilon^{2m}+\varsigma_{3m}\upsilon^{3m}+\cdots.
\end{equation}
(15)
Clearly,
\begin{equation*}
[\tau(z)]^{\chi}=1+\chi\tau_{m}z^{m}+\left(\chi\tau_{2m}+\frac{\chi(\chi-1)}{2}\tau_{m}^2\right)z^{2m}+\cdots,
\end{equation*}
and
\begin{equation*}
[\varsigma(\upsilon)]^{\chi}=1+\chi\varsigma_{m}\upsilon^{m}+\left(\chi\varsigma_{2m}+\frac{\chi(\chi-1)}{2}\varsigma_{m}^2\right)\upsilon^{2m}+\cdots.
\end{equation*}
Also
\begin{equation*}
(\mathcal{D}_{q}f(z))^{\sigma}=1+\sigma[m+1]_q\rho_{m+1}z^{m}+\left(\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\right)z^{2m}+\cdots,
\end{equation*}
and
\begin{align*}
(\mathcal{D}_{q}g(\upsilon))^{\sigma}=&1-\sigma[m+1]_q\rho_{m+1}\upsilon^{m}-\sigma[2m+1]_q\rho_{2m+1}\upsilon^{2m}\\&+\Biggl(\sigma(m+1)[2m+1]_q\rho_{m+1}^2+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\Biggr)\upsilon^{2m}+\cdots
\end{align*}
Comparing the coefficients in (12) and (13), we have
\begin{align}
\label{2.8}
&\sigma[m+1]_q\rho_{m+1}=\chi\tau_{m},
\end{align}
(16)
\begin{align}
\label{2.9}
&\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2=\chi\tau_{2m}+\frac{\chi(\chi-1)}{2}\tau_{m}^2,
\end{align}
(17)
\begin{align}
\label{2.10}
-&\sigma[m+1]_q\rho_{m+1}=\chi\varsigma_{m},
\end{align}
(18)
\begin{align}
\label{2.11}
-&\sigma[2m+1]_q\rho_{2m+1}+\Biggl(\sigma(m+1)[2m+1]_q+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\Biggr)\rho_{m+1}^2=\chi\varsigma_{2m}+\frac{\chi(\chi-1)}{2}\varsigma_{m}^2.
\end{align}
(19)
From (16) and (18), we obtain
\begin{equation}
\label{2.12}
\tau_{m}=-\varsigma_{m},
\end{equation}
(20)
and
\begin{equation}
\label{2.13}
2\sigma[m+1]_q^2\rho_{m+1}^2=\chi^2(\tau_{m}^2+\varsigma_{m}^2).
\end{equation}
(21)
Further from (17), (19) and (21), we obtain that
\begin{equation*}
\sigma(\sigma-1)\chi[m+1]_q^2\rho_{m+1}^2+(m+1)\sigma\chi[2m+1]_q\rho_{m+1}^2-(\chi-1)\sigma^2[m+1]_q^2\rho_{m+1}^2=\chi^2(\tau_{2m}+\varsigma_{2m}).
\end{equation*}
Therefore, we have
\begin{equation}
\label{2.14}
\rho_{m+1}^2=\frac{\chi^2(\tau_{2m}+\varsigma_{2m})}{\sigma[m+1]_q^2(\sigma-\chi)+(m+1)\sigma\chi[2m+1]_q}.
\end{equation}
(22)
By applying Lemma 1 for the coefficients \(\tau_{2m}\) and \(\varsigma_{2m}\), then we have
\begin{equation*}
|\rho_{m+1}|\leq\frac{2\chi}{\sqrt{(m+1)\sigma\chi[2m+1]_q-(\chi-\sigma)\sigma[m+1]_q^2}}.
\end{equation*}
Also, to find the bound on \(|\rho_{2m+1}|\), using the relation (19) and (17), we obtain
\begin{equation}
\label{2.15}
2\sigma[2m+1]_q\rho_{2m+1}-(m+1)\sigma[2m+1]_q\rho_{m+1}^2=\chi(\tau_{2m}-\varsigma_{2m})+\frac{\chi(\chi-1)}{2}(\tau_{m}^2-\varsigma_{m}^2).
\end{equation}
(23)
It follows from (20), (21) and (23),
\begin{equation}
\rho_{2m+1}=\frac{(m+1)\chi^2\tau_{m}^2}{2\sigma^2[m+1]_q^2}+\frac{\chi(\tau_{2m}-\varsigma_{2m})}{2\sigma[2m+1]_q}.
\end{equation}
(24)
Applying Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), then we have
\begin{equation*}
|\rho_{2m+1}|\leq\frac{2\chi}{\sigma[2m+1]_q}+\frac{2(m+1)\chi^2}{\sigma^2[m+1]_q^2}.
\end{equation*}
Choosing \(q\longrightarrow1^{-1}\) in Theorem 1, we get the following result:
Corollary 1.
Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma,m}(\chi)\), \((m\in \mathcal{N}, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (3). Then
\begin{equation}
|\rho_{m+1}|\leq\frac{2\chi}{\sqrt{(m+1)[\sigma\chi m+\sigma^2m+\sigma^2]}},
\end{equation}
(25)
and
\begin{equation}
|\rho_{2m+1}|\leq\frac{2\chi}{\sigma(2m+1)}+\frac{2\chi^2}{\sigma^2(m+1)}.
\end{equation}
(26)
Choosing \(m=1\) (0ne-fold case) in Theorem 1, we get the following result:
Corollary 2.
Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma}(\chi)\), \((0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (1). Then
\begin{equation}
|\rho_{2}|\leq\frac{2\chi}{\sqrt{2\sigma\chi[3]_q-(\chi-\sigma)\sigma[2]_q^2}},
\end{equation}
(27)
and
\begin{equation}
|\rho_{3}|\leq\frac{2\chi}{\sigma[3]_q}+\frac{4\chi^2}{\sigma^2[2]_q^2},
\end{equation}
(28)
Choosing \(q\longrightarrow1^{-1}\) in Corollary 2, we get the following result:
Corollary 3. [29]
Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\chi)\), \(( \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (1). Then
\begin{equation}
|\rho_{2}|\leq\frac{2\chi}{\sqrt{2\sigma(2\sigma+\chi)}},
\end{equation}
(29)
and
\begin{equation}
|\rho_{3}|\leq\frac{\chi(2\sigma+3\chi)}{3\sigma^2}.
\end{equation}
(30)
Remark 3.
For one-fold case, we have \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{q,1}_{\varSigma,1}(\chi)=\mathcal{M}_{\varSigma}(\chi)\), and we can get the results of Srivastava et al., [11].
Definition 5.
A function \(f(z)\) given by (3) is said to be in the class \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\;(m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) if
\begin{equation}
\label{3.1}
f\in\varSigma\quad and \quad \Re[(\mathcal{D}_{q}f(z))^{\sigma}]>\psi,
\end{equation}
(31)
and
\begin{equation}
\label{3.2}
\Re[(\mathcal{D}_{q}g(\upsilon))^{\sigma}]>\psi,
\end{equation}
(32)
where \(g(\upsilon)\) is given by (2).
Remark 4.
We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{\sigma}_{\varSigma,1}(\psi)=\mathcal{M}^{\sigma}_{\varSigma}(\chi)\) which was introduced and studied by Girgaonkar et al., [29].
Remark 5.
We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{1}_{\varSigma,1}(\psi)=\mathcal{M}_{\varSigma}(\chi)\) which was introduced and studied by Srivastava et al., [11].
Theorem 2.
Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\), \((m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (3). Then
\begin{equation}
|\rho_{m+1}|\leq\min\Biggl\{\frac{2(1-\psi)}{\sigma[m+1]_q},2\sqrt{\frac{1-\psi}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}}\Biggr\},
\end{equation}
(33)
and
\begin{equation}
|\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}+\frac{2(1-\psi)}{\sigma[2m+1]_q}.
\end{equation}
(34)
Proof.
Using inequalities (31) and (32), we get
\begin{equation}
\label{3.5}
(\mathcal{D}_{q}f(z))^{\sigma}=\psi+(1-\psi)\tau(z),
\end{equation}
(35)
and
\begin{equation}
\label{3.5a}
(\mathcal{D}_{q}g(\upsilon))^{\sigma}=\psi+(1-\psi)\varsigma(\upsilon),
\end{equation}
(36)
here \(\tau(z)\) and \(\varsigma(\upsilon)\) in \(\mathcal{P}\) are given by the following series
\begin{equation*}
\tau(z)=1+\tau_mz^m+\tau_{2m}z^{2m}+\tau_{3m}z^{3m}+\cdots,
\end{equation*}
and
\begin{equation*}
\varsigma(\upsilon)=1+\varsigma_m\upsilon^m+\varsigma_{2m}\upsilon^{2m}+\varsigma_{3m}\upsilon^{3m}+\cdots.
\end{equation*}
Clearly,
\begin{equation*}
\psi+(1-\psi)\tau(z)=1+(1-\psi)\tau_{m}z^{m}+(1-\psi)\tau_{2m}z^{2m}+\cdots,
\end{equation*}
and
\begin{equation*}
\psi+(1-\psi)\varsigma(\upsilon)=1+(1-\psi)\varsigma_{m}\upsilon^{m}+(1-\psi)\varsigma_{2m}\upsilon^{2m}+\cdots.
\end{equation*}
Also
\begin{equation*}
(\mathcal{D}_{q}f(z))^{\sigma}=1+\sigma[m+1]_q\rho_{m+1}z^{m}+\left(\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\right)z^{2m}+\cdots,
\end{equation*}
and
\begin{align*}
(\mathcal{D}_{q}g(\upsilon))^{\sigma}=&1-\sigma[m+1]_q\rho_{m+1}\upsilon^{m}-\sigma[2m+1]_q\rho_{2m+1}\upsilon^{2m}\\&+\Biggl(\sigma(m+1)[2m+1]_q\rho_{m+1}^2+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\Biggr)\upsilon^{2m}+\cdots.
\end{align*}
Now comparing the coefficients in (35) and (36), we get
\begin{align}
\label{3.6}
&\sigma[m+1]_q\rho_{m+1}=(1-\psi)\tau_{m},\\
\end{align}
(37)
\begin{align}
\label{3.7}
&\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2=(1-\psi)\tau_{2m},\\
\end{align}
(38)
\begin{align}
\label{3.8}
-&\sigma[m+1]_q\rho_{m+1}=(1-\psi)\varsigma_{m},\\
\end{align}
(39)
\begin{align}
\label{3.9}
-&\sigma[2m+1]_q\rho_{2m+1}+\Biggl(\sigma(m+1)[2m+1]_q+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\Biggr)\rho_{m+1}^2=(1-\psi)\varsigma_{2m}.
\end{align}
(40)
From (37) and (39), we obtain
\begin{equation}
\label{3.10}
\tau_{m}=-\varsigma_{m},
\end{equation}
(41)
and
\begin{equation}
\label{3.11}
2\sigma[m+1]_q^2\rho_{m+1}^2=(1-\psi)^2(\tau_{m}^2+\varsigma_{m}^2).
\end{equation}
(42)
Also, from (38) and (40), we get
\begin{equation}
\label{3.12}
\sigma(\sigma-1)\chi[m+1]_q^2\rho_{m+1}^2+(m+1)\sigma[2m+1]_q\rho_{m+1}^2=(1-\psi)(\tau_{2m}+\varsigma_{2m}).
\end{equation}
(43)
Applying the Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), we find that
\begin{equation*}
|\rho_{m+1}|\leq2\sqrt{\frac{(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}}.
\end{equation*}
Also, to find the bound on \(|\rho_{2m+1}|\), using the relation (40) and (38), we obtain
\begin{equation}
-(m+1)\sigma[2m+1]_q\rho_{m+1}^2+ 2\sigma[2m+1]_q\rho_{2m+1}=(1-\psi)(\tau_{2m}-\varsigma_{2m}),
\end{equation}
(44)
or equivalently
\begin{equation}
\label{3.13}
\rho_{2m+1}=\frac{(1-\psi)(\tau_{2m}-\varsigma_{2m})}{2\sigma[2m+1]_q}+\frac{(m+1)}{2}\rho_{m+1}^2.
\end{equation}
(45)
By substituting the value of \(\rho_{m+1}^2\) from (42), we have
\begin{equation}
\rho_{2m+1}=\frac{(1-\psi)(\tau_{2m}-\varsigma_{2m})}{2\sigma[2m+1]_q}+\frac{(m+1)(1-\psi)^2(\tau_{m}^2+\varsigma_{m}^2)}{4\sigma^2[m+1]_q^2}.
\end{equation}
(46)
Applying the Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), we get
\begin{equation*}
|\rho_{2m+1}|\leq\frac{2(1-\psi)}{\sigma[2m+1]_q}+\frac{2(m+1)(1-\psi)^2}{2\sigma^2[m+1]_q^2}.
\end{equation*}
Also, by using (43) and (45), and applying Lemma 1 we obtain
\begin{equation*}
|\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}+\frac{2(1-\psi)}{\sigma[2m+1]_q}.
\end{equation*}
This complete the proof.
Choosing \(q\longrightarrow1^{-1}\) in Theorem 2, we get the following result:
Corollary 4.
Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma,m}(\psi)\), \((m\in \mathcal{N}, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (3). Then
\begin{equation*}
|\rho_{m+1}|\leq\left \{
\begin{array}{cc}
2\sqrt{\frac{(1-\psi)}{\sigma(\sigma-1)[m+1]^2+(m+1)\sigma[2m+1]}} & 0\leq\psi\leq\frac{m}{1+2m},\\
\frac{2(1-\psi)}{\sigma[m+1]} & \frac{m}{1+2m}\leq\psi< 1,
\end{array}
\right.
\end{equation*}
and
\begin{equation*}
|\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]^2+(m+1)\sigma[2m+1]}+\frac{2(1-\psi)}{\sigma[2m+1]}.
\end{equation*}
For one fold case, Corollary 4, yields the following Corollary:
Corollary 5.
Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then
\begin{equation*}
|\rho_{2}|\leq\left \{
\begin{array}{cc}
\sqrt{\frac{2(1-\psi)}{\sigma(2\sigma+1)}} & 0\leq\psi\leq\frac{1}{3},\\
\frac{(1-\psi)}{\sigma} & \frac{1}{3}\leq\psi< 1,
\end{array}
\right.
\end{equation*}
and
\begin{equation*}
|\rho_{3}|\leq\frac{(1-\psi)(2\sigma-3\psi+3)}{3\sigma^2}.
\end{equation*}
Remark 6.
Corollary 5 gives above is the improvement of the estimates for coefficients on \(|\rho_{2}|\) and \(|\rho_{3}|\) investigated by Girgaonkar et al., [29].
Corollary 6. [29]
Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then
\begin{equation*}
|\rho_{2}|\leq\sqrt{\frac{2(1-\psi)}{\sigma(2\sigma+1)}},
\end{equation*}
and
\begin{equation*}
|\rho_{3}|\leq\frac{(1-\psi)(2\sigma-3\psi+3)}{3\sigma^2}.
\end{equation*}
Taking \(\sigma=1\) in Corollary 7, we get the following result:
Corollary 7. [11]
Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then
\begin{equation*}
|\rho_{2}|\leq\sqrt{\frac{2(1-\psi)}{3}},
\end{equation*}
and
\begin{equation*}
|\rho_{3}|\leq\frac{(1-\psi)(5-3\psi)}{3}.
\end{equation*}
3. Conclusion
In this present paper, two new subclasses indicated by \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\) and \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\) of function class of \(\mathcal{E}_m\) was obtained and worked on. Also, the estimates coefficients for \(|\rho_{m+1}|\) and \(|\rho_{2m+1}|\) of functions in these classes are determined.
Conflicts of Interest
The author declares no conflict of interest.
