Applications of Higher-Order q-Derivative to Meromorphic q-Starlike Function Related to Janowski Function

Abstract

By making use of a higher-order q-derivative operator, certain families of meromorphic q-starlike functions and meromorphic q-convex functions are introduced and studied. Several sufficient conditions and coefficient inequalities for functions in these subclasses are derived. The results presented in this article extend and generalize a number of previous results.

1. Introduction

Recently, q-analysis has fascinated scholars due to various applications in many areas of physics and mathematics. The applications of q-analysis were first considered by Jackson [1,2]. In recent years, some scholars have written a number of papers [3,4,5,6,7,8,9,10,11,12,13,14,15] associated with q-starlike functions and the Janowski functions [16]. In particular, Srivastava [17,18] pointed out some applications and mathematical explanations of q-derivatives in GFT. In this paper, we consider several families of meromorphically multivalent q-starlike functions by making use of the Janowski function and the higher-order q-derivative. Certain sufficient conditions and coefficient inequalities for functions in these subclasses are derived. Moreover, several previous results are generalized.

Let $\mathrm{\Sigma }\left(p\right)(p\in \mathbb{N})$ denote the family of p-valent analytic functions in $U^{*}=\{z:0<|z|<1\}$ which have the following form:

$$f\left(z\right)=z^{-p}+\sum _{k=1}^{\infty }{a}_{k-p}{z}^{k-p}.$$

(1)

Furthermore, we let $\mathrm{\Sigma }\left(1\right)=\mathrm{\Sigma }$.

If $f\in \mathrm{\Sigma }\left(p\right)$ satisfies the condition

$$\mathrm{Re}\left(-\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>0,$$

(2)

then f is called a meromorphic p-valent starlike function. We note the family by $M{\mathrm{\Sigma }}^{*}\left(p\right)$ and write $M{\mathrm{\Sigma }}^{*}\left(1\right)=M{\mathrm{\Sigma }}^{*}$. The class $M{\mathrm{\Sigma }}^{*}$ was studied by Pommerenke [19].

If $f\in \mathrm{\Sigma }\left(p\right)$ satisfies the condition

$$\mathrm{Re}\left(-\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{f^{\prime }\left(z\right)}\right)>0,$$

(3)

then f is called a meromorphic p-valent convex function. We note this family by $MC\left(p\right)$ and write $MC\left(1\right)=MC$.

If a function $\psi$ is analytic in $U=U^{*}\cup \left\{0\right\}$ and satisfies

$$\psi \left(z\right)=1+\sum _{k=1}^{\infty }{\psi }_k{z}^k$$

(4)

and

$$\mathrm{Re}\left(\psi \left(z\right)\right)>0,$$

then $\psi$ is said to be in the family P.

Let $\phi$ be analytic in U and $\phi \left(0\right)=1$. If $\phi$ satisfies

$$\phi \left(z\right)\prec \frac{Az+1}{Bz+1}(-1\leqq B<A\leqq 1),$$

then $\phi$ is said to be in $P[A,B]$.

In [16], Janowski studied the family $P[A,B]$ and obtained that $\phi$ is in $P[A,B]$ if

$$\phi \left(z\right)=\frac{(1-A)+(1+A)\psi \left(z\right)}{(1-B)+(1+B)\psi \left(z\right)}(\psi \in P;-1\leqq B<A\leqq 1).$$

Let f and g be analytic in U. If there exists w analytic in U with $w\left(0\right)=0$ and $|w(z)|<1$, so that $f\left(z\right)=g\left(w\right(z\left)\right),$ then we say that f is subordinate to g, written by $f\prec g$. Further, if the function g is analytic and univalent in U, then

$$f\prec g(z\in U)⟺f\left(U\right)\subset g\left(U\right)\mathrm{and}f\left(0\right)=g\left(0\right).$$

Definition 1.

Let $f\in \mathrm{\Sigma }$. Then f is called to be in $M{\mathrm{\Sigma }}^{*}[A,B]$ if

$$-\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}=\frac{(1-A)+(1+A)\psi \left(z\right)}{(1-B)+(1+B)\psi \left(z\right)}(-1\leqq B<A\leqq 1;\psi \in P).$$

(5)

In [20], Karunakaran studied the family $M{\mathrm{\Sigma }}^{*}[A,B]$.

Let $0<q<1$ and define ${\left[\tau \right]}_q$ as the following:

$${\left[\tau \right]}_q=\left\{\begin{array}{cc}{\displaystyle \frac{q^{\tau }-1}{q-1}}\hfill & (\tau \in \mathbb{C})\hfill \\ \sum _{k=0}^{n-1}{q}^k=1+q+q^2+\cdots +q^{n-1}\hfill & (\tau =n\in \mathbb{N}).\hfill \end{array}\right.$$

Let $0<q<1$. The q-factorial ${\left[m\right]}_q!$ is defined by

$${\left[m\right]}_q!=\left\{\begin{array}{cc}1\hfill & (m=0)\hfill \\ {\Pi }_{k=1}^m{\left[k\right]}_q\hfill & (m\in \mathbb{N}).\hfill \end{array}\right.$$

Let $\gamma \in \mathbb{N}$. We define q-Pochhammer symbol ${\left[\gamma \right]}_{q,n}$ by (see [21])

$${\left[\gamma \right]}_{q,n}=\left\{\begin{array}{cc}1\hfill & (n=0)\hfill \\ {\Pi }_{k=\gamma }^{\gamma +n-1}{\left[k\right]}_q\hfill & (n\in \mathbb{N}).\hfill \end{array}\right.$$

Specifically, we write ${\left[0\right]}_{q,n}=0$.

Next, we define the q-derivative $D_q$$(0<q<1)$ for $f\in \mathrm{\Sigma }\left(p\right)$ by

$$\begin{array}{cc}\hfill \left(D_{q}f\right)\left(z\right)& =\frac{f\left(zq\right)-f\left(z\right)}{z(q-1)}\hfill \\ \hfill & =-\sum _{k=1}^p\left(\frac{{\left[k\right]}_{q,1}}{q^k}\right)a_{-k}{z}^{-1-k}+\sum _{k=1}^{\infty }{\left[k\right]}_q{a}_k{z}^{-1+k},\hfill \end{array}$$

(6)

where $a_{-p}=1$.

From (6), we can see that

$$\underset{q\to 1}{lim}\left(D_{q}f\right)\left(z\right)=\underset{q\to 1}{lim}\frac{f\left(zq\right)-f\left(z\right)}{z(q-1)}=f^{\prime }\left(z\right).$$

Further, one can find that

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

(7)

$$\left(D_q^{\left(3\right)}f\right)\left(z\right)=-\sum _{k=1}^p\left(\frac{{\left[k\right]}_{q,3}}{q^{3k+3}}\right)a_{-k}{z}^{-k-3}+\sum _{k=3}^{\infty }{[k-2]}_{q,3}{a}_k{z}^{k-3},$$

(8)

$$\cdots \cdots \cdots$$

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

(9)

where $a_{-p}=1$ and $D_q^{\left(p\right)}$ is called pth order q-derivatives.

Definition 2.

Let $f\in \mathrm{\Sigma }$. If f satisfies

$$\left|\frac{qz{D}_{q}f\left(z\right)}{f\left(z\right)}+\frac{1}{1-q}\right|<\frac{1}{1-q},$$

(10)

then f is called to be in the meromorphic q-starlike function family $M{\mathrm{\Sigma }}_q^{*}$.

It is easily seen that, when $q\to 1^{-}$, the disk given by (10) becomes

$$\mathrm{Re}\left(-\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>0.$$

Thus, the class $M{\mathrm{\Sigma }}_q^{*}$ reduces to the meromorphic starlike function family $M{\mathrm{\Sigma }}^{*}$ (see [19]).

Furthermore, we can rewrite (10) as the following:

$$-\frac{qz{D}_{q}f\left(z\right)}{f\left(z\right)}\prec \widehat{h}\left(z\right)\mathrm{where}\widehat{h}\left(z\right)=\frac{z+1}{1-qz}.$$

Further, the meromorphic q-convex function family $M{C}_q$ could be derived by

$$f\left(z\right)\in M{C}_q\iff -qz{D}_{q}f\left(z\right)\in M{\mathrm{\Sigma }}_q^{*}.$$

Definition 3.

If a function $f\in \mathrm{\Sigma }\left(p\right)$ satisfies

$$-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}f\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}f\right)\left(z\right)}\prec \frac{(A+1)\widehat{h}\left(z\right)+(1-A)}{(B+1)\widehat{h}\left(z\right)+(1-B)}\left(\widehat{h}\left(z\right)=\frac{z+1}{1-qz}-1;\leqq B<A\leqq 1\right),$$

or, equivalently,

$$-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}f\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}f\right)\left(z\right)}\prec s\left(z\right),$$

(11)

where

$$s\left(z\right)=\frac{z(1+A)-zq(1-A)+2}{z(1+B)-zq(1-B)+2}(0<q<1;-1\leqq B<A\leqq 1),$$

(12)

then f is in the family $M{\mathrm{\Sigma }}_q^{*}[p,A,B]$.

Remark 1.

We write the following special cases:

(i) 

$M{\mathrm{\Sigma }}_q^{*}[1,A,B]=M{\mathrm{\Sigma }}_q^{*}[A,B]$, when $p=1$.

(ii) 

${lim}_{q\to 1^{-}}M{\mathrm{\Sigma }}_q^{*}[1,A,B]=M{\mathrm{\Sigma }}^{*}[A,B]$, when $p=1$.

(iii) 

$M{\mathrm{\Sigma }}_q^{*}[p,A,B]=M{\mathrm{\Sigma }}_q^{*}\left[\alpha \right]$, when $p=1$, $A=1-2\alpha$$(0\leqq \alpha <1)$ and $B=-1$. In [19], Pommerenke considered the family $M{\mathrm{\Sigma }}_q^{*}\left[\alpha \right]$.

Now we define the meromorphic q-convex function family $M{C}_q[p,A,B]$ by

$$f\in M{C}_q[p,A,B]⟺\frac{{(-1)}^p{q}^{\frac{1}{2}p(3p-1)}}{{\left[p\right]}_{q,p}}{z}^p{D}_q^{\left(p\right)}f\in M{\mathrm{\Sigma }}_q^{*}[p,A,B].$$

In particular, we write $M{C}_q[p,A,B]=M{C}_q[A,B]$ when $p=1$.

Lemma 1

([22]). Let $\psi \left(z\right)=1+{\psi }_{1}z+{\psi }_2{z}^2+\cdots$ belong to the family P. Then

$$|{\psi }_2-\nu {\psi }_1^2|\leqq \left\{\begin{array}{cc}4\nu -2\hfill & (\nu >1)\hfill \\ 2\hfill & (0\leqq \nu \leqq 1)\hfill \\ -4\nu +2\hfill & (\nu <0).\hfill \end{array}\right.$$

(13)

Lemma 2

([23]). Let $h\left(z\right)=1+{\sum }_{k=1}^{\infty }{h}_k{z}^k$ be analytic in U. Furthermore, let $H\left(z\right)=1+{\sum }_{k=1}^{\infty }{C}_k{z}^k$ be univalent convex in U. If $h\left(z\right)\prec H\left(z\right)$, then

$$|h_k|\leqq |C_1|(k\geqq 1).$$

2. Main Results

Theorem 1.

If

$$g\left(z\right)=z^{-p}+\sum _{k=1}^{\infty }{a}_{-p+k}{z}^{-p+k}\in M{\mathrm{\Sigma }}_q^{*}[p,A,B](p\geqq 2),$$

then

$$|a_{2-p}-\mu a_{1-p}^2|\leqq \left\{\begin{array}{cc}\left(\frac{A-B}{2}\right)\left(\frac{{[2p-3]}_{q,3}}{q^{2p-2}{[p-2]}_{q,2}}\right)\mathrm{\Lambda }\left(q\right)\hfill & (\mu >{\sigma }_1)\hfill \\ \left(\frac{A-B}{2}\right)\left(\frac{{[2p-3]}_{q,3}}{q^{2p-2}{[p-2]}_{q,2}}\right)\hfill & ({\sigma }_2\leqq \mu \leqq {\sigma }_1)\hfill \\ \left(\frac{B-A}{2}\right)\left(\frac{{[2p-3]}_{q,3}}{q^{2p-2}{[p-2]}_{q,2}}\right)\mathrm{\Lambda }\left(q\right)\hfill & (\mu <{\sigma }_2),\hfill \end{array}\right.$$

where

$$\mathrm{\Lambda }\left(q\right)=\frac{\left\{\begin{array}{c}\left\{(1+Bq{[2p-2]}_q-A{[2p-1]}_q)(q+1)-2\right\}{[p-1]}_q{[2p-3]}_q\hfill \\ +\mu {(q+1)}^2(A-B){[2p-2]}_{q,2}{[p-2]}_q\hfill \end{array}\right\}}{2{[p-1]}_q{[2p-3]}_q},$$

$${\sigma }_1=\frac{{[p-1]}_q{[2p-3]}_q\{4+(q+1)(A{[2p-1]}_q-qB{[2p-2]}_q-1)\}}{{(q+1)}^2(A-B){[p-2]}_q{[2p-2]}_{q,2}}$$

and

$${\sigma }_2=\frac{{[p-1]}_q{[2p-3]}_q\left\{(q+1)(A{[2p-1]}_q-1-qB{[2p-2]}_q)\right\}}{4{(q+1)}^2(A-B){[p-2]}_q{[2p-2]}_{q,2}}.$$

Proof. 

From the assumption of the theorem, we obtain

$$-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}\prec \varphi \left(z\right),$$

where

$$\varphi \left(z\right)=\frac{2-q(1-A)z+(1+A)z}{2-q(1-B)z+(1+B)z}.$$

This gives that

$$-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}=\varphi \left(w\left(z\right)\right),$$

where $w\left(z\right)$ is a Schwarz function. Now a function $h\left(z\right)$ is defined as follows:

$$h\left(z\right)=\frac{1+w\left(z\right)}{1-w\left(z\right)}=1+\sum _{n=1}^{\infty }{h}_n{z}^n\in P.$$

□

Furthermore, one can see that

$$\begin{array}{cc}\hfill & \varphi \left(w\left(z\right)\right)=\frac{2\left(h\right(z)+1-q(h\left(z\right)-1\left)\right)+(1+A)(q+1)\left(h\right(z)-1)}{2\left(h\right(z)+1-q(h\left(z\right)-1\left)\right)+(1+B)(q+1)\left(h\right(z)-1)}\hfill \\ \hfill & =1+\frac{1}{4}(1+q)(A-B)h_{1}z+\frac{1}{16}(1+q)(A-B)\left\{4h_2-(B(1+q)-q+3)h_1^2\right\}{z}^2+\cdots .\hfill \end{array}$$

Similarly, we find that

$$\begin{array}{cc}\hfill -\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}& =1-\frac{{[p-1]}_q}{{[2p-2]}_{q,2}}{q}^{p-1}{a}_{1-p}z\hfill \\ \hfill & +\frac{{[p-1]}_q}{{\left[2(p-1)\right]}_{q,2}}{q}^{2(p-1)}\left(\frac{{[p-1]}_q}{{\left[2(p-1)\right]}_q}{a}_{1-p}^2-\frac{{[p-2]}_q}{{[2p-3]}_q}(1+q)a_{2-p}\right)z^2+\cdots \hfill \end{array}$$

for $p\geqq 2$. Therefore, for $p\geqq 2$, we obtain

$$a_{1-p}=-\frac{(q+1)(A-B){\left[2(p-1)\right]}_{q,2}}{4q^{p-1}{[p-1]}_q}{h}_1$$

(14)

and

$$a_{2-p}=\frac{(A-B){[2p-3]}_{q,3}}{q^{2p-2}{[p-2]}_{q,2}}\left(\frac{1}{16}{k}_1\left(q\right)h_1^2-\frac{1}{4}{h}_2\right),$$

(15)

where

$$k_1\left(q\right)=(1+q)\{A{[2p-1]}_q-B({[2p-1]}_q-1)-1\}+4.$$

(16)

Hence we obtain for $p\geqq 2$ that

$$|a_{2-p}-\mu a_{1-p}^2|=\left(\frac{A-B}{4}\right)\left(\frac{{[2p-3]}_{q,3}}{q^{2p-2}{[p-2]}_{q,2}}\right)|h_2-k_2{h}_1^2|,$$

(17)

where

$$k_2=\frac{{[p-1]}_q{[2p-3]}_q{k}_1\left(q\right)-\mu {\left[2(p-1)\right]}_{q,2}{[p-2]}_q(A-B){(q+1)}^2}{4{[2p-3]}_q{[p-1]}_q}$$

with $k_1\left(q\right)$ given by (16).

Now we can see that the conditions $\mu >{\sigma }_1$, ${\sigma }_2\leqq \mu \leqq {\sigma }_1$ and $\mu <{\sigma }_2$ in Theorem 1 imply that $k_2<0$, $0\leqq k_2\leqq 1$ and $k_2>1$, respectively. By applying Lemma 1 in (17), the desired result is obtained. This proves Theorem 1.

Applying the same method as in the proof of Theorem 1, we obtain the following theorem for the case $p=1$.

Theorem 2.

If

$$g\left(z\right)=z^{-1}+\sum _{k=1}^{\infty }{a}_{k-1}{z}^{k-1}\in M{\mathrm{\Sigma }}_q^{*}[A,B],$$

then

$$|a_1-\mu a_0^2|\leqq \left\{\begin{array}{cc}\left(\frac{A-B}{4}\right)[(1-A)q+\mu {(q+1)}^2(A-B)-A-1]\hfill & \left(\mu >\frac{(A-1)q+A+3}{(A-B){(q+1)}^2}\right)\hfill \\ \frac{A-B}{2}\hfill & \left(\frac{A-1}{(A-B)(q+1)}\leqq \mu \leqq \frac{(A-1)q+A+3}{(A-B){(q+1)}^2}\right)\hfill \\ \left(\frac{B-A}{4}\right)[(1-A)q+\mu {(q+1)}^2(A-B)-A-1]\hfill & \left(\mu <\frac{A-1}{(A-B)(q+1)}\right).\hfill \end{array}\right.$$

Letting $q\to 1^{-}$, $A=1$ and $B=-1$ in Theorem 2, we obtain a result of the known family $M{\mathrm{\Sigma }}^{*}$.

Corollary 1.

If

$$g\left(z\right)=z^{-1}+\sum _{k=1}^{\infty }{a}_{k-1}{z}^{k-1}\in M{\mathrm{\Sigma }}^{*},$$

then

$$\left.a_1-\mu a_0^2\right|\leqq \left\{\begin{array}{cc}4\mu -1\hfill & (\mu >\frac{1}{2})\hfill \\ 1\hfill & (0\leqq \mu \leqq \frac{1}{2})\hfill \\ 1-4\mu \hfill & (\mu <0).\hfill \end{array}\right.$$

Theorem 3.

Let $p\geqq 2$. If

$$g\left(z\right)=z^{-p}+\sum _{k=1}^{\infty }{a}_{-p+k}{z}^{-p+k}\in M{\mathrm{\Sigma }}_q^{*}[p,A,B],$$

then

$$\left|a_{-p+k}\right|\leqq \prod _{j=1}^k\frac{2{[j-1]}_q+{[2p-1]}_q(q+1)(A-B)}{2{\left[j\right]}_q{q}^{p-1}{[p-j]}_q}$$

(18)

for $1\leqq k\leqq p-1$.

Proof. 

If g belongs to $M{\mathrm{\Sigma }}_q^{*}[p,A,B]$, then

$$\psi \left(z\right):=-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}\prec \varphi \left(z\right),$$

(19)

where

$$\varphi \left(z\right)=\frac{z(1+A)-zq(1-A)+2}{z(1+B)-zq(1-B)+2}=1-\frac{1}{2}(1+q)(B-A)z+\frac{1}{4}(1+q)(B-A)\left\{1+B(1+q)-q\right\}{z}^2+\cdots .$$

□

Let

$$\psi \left(z\right)=1+\sum _{k=1}^{\infty }{\psi }_k{z}^k.$$

Applying Lemma 2, we obtain

$$|{\psi }_k|\leqq \frac{1}{2}(q+1)(A-B)(k\geqq 1).$$

(20)

Furthermore, from (19), we have

$$-q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)=\left\{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)\right\}\psi \left(z\right),$$

which implies that

$$\begin{array}{cc}\hfill & -q^{2p-1}\left(\sum _{k=1}^p\frac{{(-1)}^p{\left[k\right]}_{q,p}}{q^{p\left[k+\frac{1}{2}(p-1)\right]}}{a}_{-k}{z}^{-p-k+1}+\sum _{k=p}^{\infty }{[-p+k+1]}_{q,p}{a}_k{z}^{-p+k+1}\right)\hfill \\ \hfill & ={[2p-1]}_q\left(1+\sum _{k=1}^{\infty }{\psi }_k{z}^k\right)\left(\sum _{k=1}^p\frac{{(-1)}^{p-1}{\left[k\right]}_{q,p-1}}{q^{\left[k+\frac{1}{2}(p-2)\right](p-1)}}{a}_{-k}{z}^{-p-k+1}\right.\hfill \\ \hfill & \left.+\sum _{k=p-1}^{\infty }{[2+k-p]}_{q,p-1}{a}_k{z}^{-p+k+1}\right),\hfill \end{array}$$

where $a_{-p}=1$.

It is easily seen from the above formula that

$$|a_{-p+k}|\leqq \frac{(A-B)(q+1){[2p-1]}_q{q}^{\frac{1}{2}(p-1)(3p-2k-2)}}{2{\left[k\right]}_q{[p-k]}_{q,p-1}}\sum _{l=1}^k\frac{{[p+l-k]}_{q,p-1}}{q^{\frac{1}{2}(3p-2k+2l-2)(p-1)}}\left|a_{k-p-l}\right|$$

(21)

for $1\leqq k\leqq p-1$.

Now

$$\begin{array}{cc}\hfill & |a_{1-p}|\leqq \frac{{[2p-2]}_{q,2}(1+q)(A-B)}{2q^{p-1}{[p-1]}_q},\hfill \\ \hfill & |a_{2-p}|\leqq \frac{q^{\frac{1}{2}(p-1)(3p-6)}{[2p-1]}_q(1+q)(A-B)}{2{[p-2]}_{q,p-1}{\left[2\right]}_q}\left\{\frac{{\left[p\right]}_{q,p-1}}{q^{\frac{1}{2}(3p-2)(p-1)}}+\frac{{[p-1]}_{q,p-1}}{q^{\frac{1}{2}(3p-4)(p-1)}}|a_{1-p}|\right\}\hfill \\ \hfill & =\frac{(1+q)(A-B){[2p-2]}_{q,2}}{2q^{p-1}{[p-1]}_q}·\frac{\{2+{[2p-1]}_q(1+q)(A-B)\}{[2p-3]}_q}{2q^{p-1}{\left[2\right]}_q{[p-2]}_q},\hfill \\ \hfill & \cdots \cdots \cdots \hfill \\ \hfill & |a_{k-p}|\leqq \prod _{j=1}^k\frac{2{[j-1]}_q+(1+q)(A-B){[2p-1]}_q}{2q^{p-1}{\left[j\right]}_q{[p-j]}_q}\hfill \end{array}$$

for $1\leqq k\leqq p-1$. This proves Theorem 3.

Applying the same methods as in the proof of Theorem 3, we obtain the following Theorems 4 and 5.

Theorem 4.

Let $p\geqq 2$. If $g\left(z\right)=z^{-p}+{\sum }_{k=1}^{\infty }{a}_{-p+k}{z}^{-p+k}$ belongs to $M{C}_q[p,A,B],$ then

$$|a_{-p+k}|\leqq \frac{{\left[p\right]}_{q,p}}{q^{pk}{[p-k]}_{q,p}}\prod _{n=1}^k\frac{{[2p-1]}_q(1+q)(A-B)+2{[n-1]}_q}{2{\left[n\right]}_q{q}^{p-1}{[p-n]}_q}$$

for $1\leqq k\leqq p-1$.

Theorem 5.

Let $g\left(z\right)=z^{-1}+{\sum }_{k=1}^{\infty }{a}_{k-1}{z}^{k-1}$ belong to $M{\mathrm{\Sigma }}_q^{*}[A,B].$ Then

$$|a_{k-1}|\leqq \prod _{n=1}^k\frac{(1+q)(A-B)+2{[n-1]}_q}{2{\left[n\right]}_q}$$

for $k\geqq 2$.

Letting $q\to 1-$, $A=1-2\alpha$$(0\leqq \alpha <1)$ and $B=-1$ in Theorem 5, we have a result of the known family $M{\mathrm{\Sigma }}^{*}\left(\alpha \right)$.

Corollary 2.

Let $g\left(z\right)=z^{-1}+{\sum }_{k=1}^{\infty }{a}_{k-1}{z}^{k-1}\in M{\mathrm{\Sigma }}^{*}\left(\alpha \right).$ Then

$$|a_{k-1}|\leqq \prod _{j=1}^k\frac{j-2\alpha +1}{j}$$

for $k\geqq 2$.

The following equivalence could help us to study the family $M{\mathrm{\Sigma }}_q^{*}[p,A,B]$:

$$g\in M{\mathrm{\Sigma }}_q^{*}[p,A,B]⟺\left|\frac{(1-B)\left\{\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}\right\}+(1-A)}{(1+B)\left(-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}\right)-(1+A)}-\frac{1}{1-q}\right|<\frac{1}{1-q}.$$

Theorem 6.

If a function $g\left(z\right)=z^{-p}+{\sum }_{k=1}^{\infty }{a}_{-p+k}{z}^{-p+k}\in \mathrm{\Sigma }\left(p\right)$ satisfies

$$\begin{array}{cc}\hfill & \sum _{k=1}^p\left(\frac{{\left[k\right]}_{q,p-1}}{q^{\left[k+\frac{1}{2}(p-2)\right](p-1)}}\right)\left\{\left|{[2p-1]}_q(1+A)-{[p+k-1]}_q{q}^{p-k}(1+B)\right|+2{[p-k]}_q\right\}|a_{-k}|\hfill \\ \hfill & +\sum _{k=p}^{\infty }{[k-p]}_{q,p-1}\left\{\left|{[2p-1]}_q(1+A)-{[p+k-1]}_q{q}^{2p-1}(1+B)\right|+2{[k+p]}_q\right\}|a_k|\hfill \\ \hfill & +{[2p-1]}_q{[p-1]}_q!(2+A)|a_{p-1}|<\frac{(A-B){\left[p\right]}_{q,p}}{q^{\frac{1}{2}(3p-2)(p-1)}},\hfill \end{array}$$

(22)

then $g\in M{\mathrm{\Sigma }}_q^{*}[p,A,B]$.

Proof. 

By a simple calculation, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{(1-B)\left\{\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}\right\}+(1-A)}{(1+B)\left\{-\frac{q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)}{{[2p-1]}_q\left(D_q^{(p-1)}g\right)\left(z\right)}\right\}-(1+A)}-\frac{1}{1-q}\right|\hfill \\ \hfill & \leqq \left|\frac{\{-q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)\}(1-B)-{[2p-1]}_q(1-A)\left(D_q^{(p-1)}g\right)\left(z\right)}{\{-q^{2p-1}z\left(D_q^{\left(p\right)}g\right)\left(z\right)\}(1+B)-{[2p-1]}_q(1+A)\left(D_q^{(p-1)}g\right)\left(z\right)}+1\right|+\frac{q}{1-q}\hfill \\ \hfill & =2\left|\frac{\left\{\begin{array}{c}{\sum }_{k=1}^p\frac{{(-1)}^{p-1}{\left[k\right]}_{q,p-1}{[2p-1]}_q}{q^{(p-1)\left[k+\frac{1}{2}(p-2)\right]}}{a}_{-k}{z}^{p-k}+{\sum }_{k=p-1}^{\infty }{[k+2-p]}_{q,p}{[2p-1]}_q{a}_k{z}^{k+p}\hfill \\ +{\sum }_{k=1}^p\frac{{(-1)}^p{\left[k\right]}_{q,p}{q}^{2p-1}}{q^{p\left[k+\frac{1}{2}(p-1)\right]}}{a}_{-k}{z}^{p-k}+{\sum }_{k=p}^{\infty }{q}^{2p-1}{[k+1-p]}_{q,p}{a}_k{z}^{n+p}\hfill \end{array}\right\}}{\left\{\begin{array}{c}(A-B)\frac{{(-1)}^{p-1}{\left[p\right]}_{q,p}}{q^{\frac{1}{2}(p-1)(3p-2)}}+(A+1){[p-1]}_q!{[2p-1]}_q{a}_{p-1}{z}^{2p-1}\hfill \\ +{\sum }_{k=1}^{p-1}\frac{{(-1)}^{p-1}{\left[k\right]}_{q,p-1}}{q^{(p-1)\left[k+\frac{1}{2}(p-2)\right]}}\{{[2p-1]}_q(A+1)-(1+B)q^{p-k}{[p-1+k]}_q\}{a}_{-k}{z}^{p-k}\hfill \\ +{\sum }_{k=p}^{\infty }\{{[2p-1]}_q(A+1)-(B+1)q^{2p-1}{[p-1+k]}_q\}{[k-p]}_{q,p+1}{a}_k{z}^{k+p}\hfill \end{array}\right\}}\right|\hfill \\ \hfill & +\frac{q}{1-q}\hfill \\ \hfill & \leqq 2\frac{{\sum }_{k=1}^{p-1}\frac{{\left[k\right]}_{q,p-1}{[p-k]}_q}{q^{(p-1)\left[k+\frac{1}{2}(p-2)\right]}}|a_{-k}{|+[p-1]}_q!{[2p-1]}_q|a_{p-1}|+{\sum }_{k=p}^{\infty }{[k-p]}_{q,p+1}{[k+p]}_q|a_k|}{\left\{\begin{array}{c}\left(\frac{{\left[p\right]}_{q,p}}{q^{\frac{1}{2}(p-1)(3p-2)}}(A-B)-(1+A){[p-1]}_q!{[2p-1]}_q|a_{p-1}|\right)\hfill \\ -{\sum }_{k=1}^{p-1}\frac{{\left[k\right]}_{q,p-1}}{q^{\left[k+\frac{1}{2}(p-2)\right](p-1)}}\left|\left\{{[2p-1]}_q(A+1)-(B+1){[p+k-1+]}_q{q}^{p-k}\right\}\right||a_{-k}|\hfill \\ -{\sum }_{k=p}^{\infty }{[k-p]}_{q,p+1}\left|\left\{{[2p-1]}_q(A+1)-(B+1)q^{2p-1}{[p+k-1]}_q\right\}\right||a_k|\hfill \end{array}\right\}}+\frac{q}{1-q}.\hfill \end{array}$$

(23)

□

Now, it follows from (22) that the last expression in (23) is less than $\frac{1}{1-q}$. This proves Theorem 6.

Theorem 7.

If a function $g\left(z\right)=z^{-1}+{\sum }_{k=1}^{\infty }{a}_{k-1}{z}^{k-1}\in \mathrm{\Sigma }$ satisfies

$$\sum _{k=1}^{\infty }\left\{2{\left[k\right]}_q+\left|q{[k-1]}_q(1+B)-(1-A)\right|\right\}|a_{k-1}|<A-B,$$

(24)

then $g\in M{\mathrm{\Sigma }}_q^{*}[A,B]$.

Proof. 

By simple calculation, we have

$$\begin{array}{cc}\hfill & \left|\frac{(1-B)\left(\frac{qz\left(D_{q}g\right)\left(z\right)}{g\left(z\right)}\right)+(1-A)}{(1+B)\left(-\frac{qz\left(D_{q}g\right)\left(z\right)}{g\left(z\right)}\right)-(1+A)}-\frac{1}{1-q}\right|\hfill \\ \hfill & \leqq \frac{q}{1-q}+2\left|\frac{g\left(z\right)+qz\left(D_{q}g\right)\left(z\right)}{q(1+B)z\left(D_{q}g\right)\left(z\right)+(1+A)}\right|\hfill \\ \hfill & =\frac{q}{1-q}+2\left|\frac{{\sum }_{k=1}^{\infty }{\left[k\right]}_q|a_{k-1}|}{(A-B)-{\sum }_{k=1}^{\infty }[q(1+B){[k-1]}_q+(1+A)]|a_{k-1}|}\right|.\hfill \end{array}$$

(25)

From (24) we can see that (25) is less than $\frac{1}{1-q}$. This proves Theorem 7. □

Letting $A=1-2\alpha$$(0\leqq \alpha <1)$, $B=-1$ and $q\to 1^{-}$ in Theorem 7, we obtain a result of the known family $M{\mathrm{\Sigma }}^{*}\left(\alpha \right)$.

Corollary 3.

If a function $g\left(z\right)=z^{-1}+{\sum }_{k=1}^{\infty }{a}_{k-1}{z}^{k-1}\in \mathrm{\Sigma }$ satisfies

$$\sum _{k=1}^{\infty }(1+k-\alpha )|a_{k-1}|<1-\alpha (0\leqq \alpha <1),$$

then $g\in M{\mathrm{\Sigma }}^{*}\left(\alpha \right)$.

Applying the same method as in the proof of Theorem 7, we obtain the following theorem.

Theorem 8.

If a function $g\left(z\right)=z^{-1}+{\sum }_{k=1}^{\infty }{a}_{k-1}{z}^{k-1}\in \mathrm{\Sigma }$ satisfies

$$\sum _{k=1}^{\infty }\left\{2{\left[k\right]}_q+\left|q{[k-1]}_q(1+B)-(1-A)\right|\right\}{[k-1]}_q|a_{k-1}|<A-B,$$

then $g\in M{C}_q[A,B].$