Applications of a q-Salagean type operator on multivalent functions

Abstract

In this paper, we introduce a new class k-$\mathcal{US}(q,\gamma ,m,p)$, $\gamma \in\mathbb{C}\backslash \{0\}$, of multivalent functions using a newly defined q-analogue of a Salagean type differential operator. We investigate the coefficient problem, Fekete–Szego inequality, and some other properties related to subordination. Relevant connections of the results presented here with those obtained in the earlier work are also pointed out.

1 Introduction

For a positive integer p, let $\mathcal{A}_{p}$ denote the set of all functions $f(z)$ which are analytic and p-valent in the open unit disk $E=\{z\in\mathbb{C}: \vert z \vert <1\}$ and have series expansion of the form

$$ f(z)=z^{p}+\sum_{n=p+1}^{\infty }a_{n}z^{n}. $$

(1.1)

Also, let $f\ast g$ denote the convolution (or Hadamard product) of $f,g\in \mathcal{A}_{p}$ defined as follows:

$$ (f\ast g) (z)=z^{p}+\sum_{n=p+1}^{\infty }a_{n}b_{n}z^{n}, $$

where $f(z)$ is given by (1.1) and $g(z)=z^{p}+\sum_{n=p+1}^{\infty }b_{n}z^{n}$.

Quite recently, q-analysis has influenced the researchers a lot due to rapid applications in mathematics and related fields. In the last century many well-known researchers (for details, see [1, 4, 6–10, 13, 14, 21, 22, 32]) did great work on q-calculus and found numerous applications. It is worth mentioning that convolution theory helps many researchers to investigate a number of properties of analytic univalent and multivalent functions. Several differential and integral operators were defined using ordinary derivative; for details, see [29].

Due to growing applications of q-calculus, investigators are interested in studying properties of functions using q-operators instead of ordinary differential operators; for comprehensive study, we refer to Kanas and Reducanu [15], Mahmood and Darus [19], and Mahmood and Sokol [20]. In this paper we define a q-analogue of a Salagean type operator and study its effect on multivalent functions in conic domains.

For any non-negative integer n, the q-integer number n denoted by $[n]_{q}$ is defined by

$$ {}[ n]_{q}=\frac{1-q^{n}}{1-q},\qquad [0]_{q}=0. $$

For a non-negative integer n, the q-number shift factorial is defined as

$$ {}[ n]_{q}!=[1]_{q}[2]_{q}[3]_{q}...[n]_{q} \quad \bigl( [0]_{q}!=1 \bigr) . $$

We note that when $q\rightarrow 1$, $[n]_{q}!$ reduces to the classical definition of factorial. In general, $[t]_{q}$ is defined as follows:

$$ {}[ t]_{q}=\frac{1-q^{t}}{1-q},\qquad [0]_{q}=0,\quad q\in ( 0,1 ) . $$

For $f\in A$, in [5], the q-derivative operator or q-difference operator is defined as follows:

$$ \partial _{q}f(z)=\frac{f(qz)-f(z)}{z(q-1)}. $$

It can easily be seen that

$$ \partial _{q}z^{n}=[n]_{q}z^{n-1},\qquad \partial _{q} \Biggl\{ \sum_{n=1}^{\infty }a_{n}z^{n} \Biggr\} =\sum_{n=1}^{\infty }[n]_{q}a_{n}z^{n-1}. $$

Taking motivation from the above mentioned work, we define new convolution operators as follows.

Let

$$ \varPhi ( p,q,m,z ) =z^{p}+\sum_{n=p+1}^{\infty } \bigl[n+ ( p-1 ) \bigr]_{q}^{m}z^{n}. $$

(1.2)

Using the functions $\varPhi ( p,q,m,z )$ and the definition of q-derivative along with the idea of convolution, we now define the following differential operator $\mathcal{S}_{q,p}^{m}f(z):\mathcal{A}_{p}\rightarrow \mathcal{A}_{p}$ for multivalent functions

$$\begin{aligned} \mathcal{S}_{q,p}^{m}f(z) =&\varPhi ( p,q,m,z ) \ast f(z),\quad m\in N\cup \{0\}, \\ =&z^{p}+\sum_{n=p+1}^{\infty }\bigl[n+ ( p-1 ) \bigr]_{q}^{m}a_{n}z^{n}, \\ =&z^{p}+\sum_{n=p+1}^{\infty }\psi _{n}a_{n}z^{n}, \end{aligned}$$

(1.3)

where

$$ \psi _{n}=\bigl[n+ ( p-1 ) \bigr]_{q}^{m}. $$

(1.4)

For $p=1$, the operator $\mathcal{S}_{q,p}^{m}f(z)$ reduces to the Salagean q-differential operator defined by Govindaraj and Sivasubramanian [11], and for $p=1$, $q\rightarrow 1$, the operator $\mathcal{S}_{q,p}^{m}f(z)$ reduces to the Salagean differential operator defined by Salagean [26].

Taking motivation from [12] and using (1.3), we define a new class k-$\mathcal{US}(q,\gamma ,m,p)$ of multivalent functions as follows.

Throughout paper we shall assume $k\geq 0$, $m\in N\cup \{0\}$, $q\in ( 0,1 )$, $\gamma \in \mathbb{C} \backslash \{0\}$, and $p\in N$.

Definition 1.1

A function $f(z)\in \mathcal{A}_{p}$ is in the class k-$\mathcal{US}(q,\gamma ,m,p)$ if it satisfies the condition

$$ \operatorname{Re} \biggl\{ 1+\frac{1}{\gamma } \biggl\{ \frac{1}{[p]_{q}} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\} \biggr\} >k \biggl\vert \frac{1}{\gamma } \biggl\{ \frac{1}{[p]_{q}} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\} \biggr\vert ,\quad z\in E. $$

By taking specific values of parameters, we obtain many important subclasses studied by various authors in earlier papers. Here we enlist some of them.

• (i) For $p=1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class k-$\mathcal{US}(q,\gamma ,m)$ studied by Saqib et al. [12].

• (ii) For $p=1$, $m=0$, $k=0$, and $\gamma \in \mathbb{C} \backslash \{0\}$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{S}_{q}^{\ast }(\gamma )$ studied by Seoudy and Aouf [27].

• (iii) For $p=1$, $m=0$, $k=0$, and $\gamma =\frac{1}{1-\alpha }$, with $0\leq \alpha <1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{S}_{q}^{\ast }(\alpha )$ studied by Agrawal and Sahoo [2].

• (iv) For $p=1$, $m=0$, $q\rightarrow 1$, and $\gamma =\frac{1}{1-\alpha }$, with $0\leq \alpha <1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{SD}(k,\alpha )$ studied by Shams et al. [28].

• (v) For $p=1$, $m=0$, $q\rightarrow 1$, and $\gamma =\frac{2}{1-\alpha }$, with $0\leq \alpha <1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{KD}(k,\alpha )$ studied by Owa et al. [24].

• (vi) For $p=1$, $k=1$, $m=0$, $q\rightarrow 1$, and $\gamma =\frac{1}{1-\alpha }$, with $0\leq \alpha <1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{S}(\alpha )$ studied by Ali et al. [3].

• (vii) For $p=1$, $k=1$, $m=0$, $q\rightarrow 1$, and $\gamma =\frac{2}{1-\alpha }$, with $0\leq \alpha <1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{K}(\alpha )$ studied by Ali et al. [3].

• (viii) For $p=1$, $m=0$, $q\rightarrow 1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{K}$-$\mathcal{ST}$ introduced by Kanas and Wisniowska [17].

• (ix) For $p=1$, $k=0$, $m=0$, $q\rightarrow 1$, and $\gamma =\frac{1}{1-\alpha }$, with $0\leq \alpha <1$, the class k-$\mathcal{US}(q,\gamma ,m,p)$ reduces to the class $\mathcal{S}^{\ast }(\alpha )$, a well-known class of starlike functions of order α, respectively.

Geometric interpretation. A function $f(z)\in \mathcal{A}_{p}$ is in the class k-$\mathcal{US}(q,\gamma ,m,p)$ if and only if $\frac{1}{[p]_{q}} ( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} )$ takes all the values in the conic domain $\varOmega _{k,\gamma }=h_{k,\gamma }(E)$ such that

$$ \varOmega _{k,\gamma }=\gamma \varOmega _{k}+(1-\alpha ), $$

where

$$ \varOmega _{k}= \bigl\{ u+iv:u>k\sqrt{ ( u-1 ) ^{2}+v^{2}} \bigr\} . $$

Since $h_{k,\gamma }(z)$ is convex univalent, so the above definition can be written as

$$ \frac{1}{[p]_{q}} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) \prec h_{k,\gamma }(z), $$

(1.5)

where

$$ h_{k,\gamma }(z)= \textstyle\begin{cases} \frac{1+z}{1-z},& \mbox{for }k=0, \\ 1+\frac{2\gamma }{\pi ^{2}} ( \log \frac{1+\sqrt{z}}{1-\sqrt{z}} ) ^{2},& \mbox{for }k=1, \\ 1+\frac{2\gamma }{1-k^{2}}\sinh ^{2} \{ ( \frac{2}{\pi }\arccos k ) \arctan h\sqrt{z} \} ,& \mbox{for }0< k< 1, \\ 1+\frac{\gamma }{k^{2}-1}\sin ( \frac{\pi }{2R(t)}\int_{0}^{\frac{u(z)}{\sqrt{t}}}\frac{1}{\sqrt{1-x^{2}}\sqrt{1-(tx)^{2}}}\,dx ) +\frac{\gamma }{1-k^{2}},& \mbox{for }k>1.\end{cases} $$

(1.6)

The boundary $\partial \varOmega _{k,\gamma }$ of the above set becomes an imaginary axis when $k=0$, and a hyperbola when $0< k<1$. For $k=1$, the boundary $\partial \varOmega _{k,\gamma }$ becomes a parabola and it is an ellipse when $k>1$ and in this case where

$$ u(z)=\frac{z-\sqrt{t}}{1-\sqrt{t}z},\quad z\in E, $$

and $t\in (0,1)$ is chosen such that $k=\cosh (\pi K^{\prime }(t)/(4K(t)))$. Here $K(t)$ is Legendre’s complete elliptic integral of the first kind and $K^{\prime }(t)=K(\sqrt{1-t^{2}})$, and $K^{\prime } ( t )$ is the complementary integral of $K ( t )$ (for details, see [16, 17, 23]). Moreover, $h_{k,\gamma }(E)$ is convex univalent in E, see [16, 17]. All of these curves have the vertex at the point $\frac{k+\gamma }{k+1}$.

2 A set of lemmas

Each of the following lemmas will be needed in our present investigation.

Lemma 2.1

([25])

Let $h(z)=\sum_{n=1}^{\infty }h_{n}z^{n}\prec F(z)=\sum_{n=1}^{\infty }d_{n}z^{n}$ in E. If $F(z)$ is convex univalent in E, then

$$ \vert h_{n} \vert \leq \vert d_{1} \vert ,\quad n \geq 1. $$

Lemma 2.2

([31])

Let $k\in {}[ 0,\infty )$ and let $h_{k,\gamma }$ be defined (1.6). If

$$\begin{aligned} &h_{k,\gamma }(z)=1+Q_{1}z+Q_{2}z^{2}+\cdots, \end{aligned}$$

(2.1)

$$\begin{aligned} &Q_{1} = \textstyle\begin{cases} \frac{2\gamma A^{2}}{1-k^{2}}, & 0\leq k< 1 ,\\ \frac{8\gamma }{\pi ^{2}}, & k=1, \\ \frac{\pi ^{2}\gamma }{4 ( 1+t ) \sqrt{t}K^{2}(t) ( k^{2}-1 ) }, & k>1,\end{cases}\displaystyle \end{aligned}$$

(2.2)

$$\begin{aligned} &Q_{2} = \textstyle\begin{cases} \frac{A^{2}+2}{3}Q_{1}, & 0\leq k< 1 ,\\ \frac{2}{3}Q_{1}, & k=1, \\ \frac{4K^{2}(t) ( t^{2}+6t+1 ) -\pi ^{2}}{24K^{2} ( t ) ( 1+t ) \sqrt{t}}Q_{1}, & k>1,\end{cases}\displaystyle \end{aligned}$$

(2.3)

where $A=\frac{2\cos ^{-1}k}{\pi }$, and $t\in (0,1)$ is chosen such that $k=\cosh ( \frac{\pi K^{\prime}(t)}{K(t)} )$, $K(t)$ is Legendre’s complete elliptic integral of the first kind.

Lemma 2.3

([18])

Let $h(z)=1+\sum_{n=1}^{\infty }c_{n}z^{n}$ be analytic in E and satisfy $\operatorname{Re}\{h(z)\}>0$ for z in E. Then the following sharp estimate holds:

$$ \bigl\vert c_{2}-\mu c_{1}^{2} \bigr\vert \leq 2\max \bigl\{ 1, \vert 2\mu -1 \vert \bigr\} ,\quad \forall \mu \in \mathbb{C} . $$

3 Main results

In this section, we will prove our main results.

Theorem 3.1

Let $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$. Then

$$ S_{q,p}^{m}f(z)\prec z\exp \int_{0}^{z}\frac{p \{ h_{k,\gamma }(w(z)) \} -1}{\zeta }\,d\xi , $$

(3.1)

where $w(z)$ is analytic in E with $w(0)=0$ and $\vert w(z) \vert <1$. Moreover, for $\vert z \vert =\rho$, we have

$$ \exp \biggl( \int_{0}^{1}\frac{p \{ h_{k,\gamma }(-\rho ) \} -1}{\rho }\,d\rho \biggr) \leq \biggl\vert \frac{S_{q,p}^{m}f(z)}{z} \biggr\vert \leq \exp \biggl( \int_{0}^{1}\frac{p \{ h_{k,\gamma }(\rho ) \} -1}{\rho }\,d\rho \biggr) , $$

(3.2)

where $h_{k,\gamma }(z)$ is defined by (1.6).

Proof

If $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$, then using identity (1.5), we obtain

$$\begin{aligned} &\frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) \prec h_{k,\gamma }(z), \\ &\frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) =h_{k,\gamma }\bigl(w(z)\bigr), \\ &\frac{\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)}-\frac{1}{z} =\frac{p \{ h_{k,\gamma }(w(z)) \} -1}{z}. \end{aligned}$$

(3.3)

For some function $w(z)$ is analytic in E with $w(0)=0$ and $\vert w(z) \vert <1$. Integrating (3.3) and after some simplification, we have

$$ S_{q,p}^{m}f(z)\prec z\exp \int_{0}^{z}\frac{p \{ h_{k,\gamma }(w(z)) \} -1}{\zeta }\,d\xi . $$

(3.4)

This proves (3.1). Noting that the univalent function $h_{k,\gamma }(z)$ maps the disk $|z|<\rho$ $(0<\rho \leq 1)$ onto a region which is convex and symmetric with respect to the real axis, we see

$$ h_{k,\gamma }\bigl(-\rho \vert z \vert \bigr)\leq \operatorname{Re} \bigl\{ h_{k,\gamma }(w(\rho z) \bigr\} \leq h_{k,\gamma }\bigl(\rho \vert z \vert \bigr)\quad (0< \rho \leq 1, z\in E). $$

(3.5)

Using (3.4) and (3.5) gives

$$\begin{aligned} \int_{0}^{1}\frac{p \{ h_{k,\gamma }(-\rho \vert z \vert ) \} -1}{\rho }\,d\rho \leq& \operatorname{Re} \int_{0}^{1}\frac{p \{ h_{k,\gamma }(w ( \rho ( z ) ) \} -1}{\rho }\,d\rho \\ \leq& \int_{0}^{1}\frac{p \{ h_{k,\gamma }(\rho \vert z \vert ) \} -1}{\rho }\,d\rho \end{aligned}$$

for $z\in E$. Consequently, subordination (3.4) leads to

$$\begin{aligned} &\int_{0}^{1}\frac{p \{ h_{k,\gamma }(-\rho \vert z \vert ) \} -1}{\rho }\,d\rho \leq \log \biggl\vert \frac{S_{q,p}^{m}f(z)}{z} \biggr\vert \leq \int_{0}^{1}\frac{p \{ h_{k,\gamma }(\rho \vert z \vert ) \} -1}{\rho }\,d\rho , \\ &h_{k,\gamma }(-\rho )\leq h_{k,\gamma }\bigl(-\rho \vert z \vert \bigr),\qquad h_{k,\gamma }\bigl(\rho \vert z \vert \bigr)\leq h_{k,\gamma}(\rho ) \end{aligned}$$

implies that

$$\begin{aligned} \exp \int_{0}^{1}\frac{p \{ h_{k,\gamma }(-\rho ) \} -1}{\rho }\,d\rho &\leq \biggl\vert \frac{S_{q,p}^{m}f(z)}{z} \biggr\vert \\ & \leq \exp \int_{0}^{1}\frac{p \{ h_{k,\gamma }(\rho ) \} -1}{\rho }\,d\rho . \end{aligned}$$

This completes the proof. □

When $p=1$, we have the following known result proved by Saqib et al. in [12].

Corollary 3.2

Let $f(z)\in k-\mathcal{US}(q,\gamma ,m)$. Then

$$ S_{q}^{m}f(z)\prec z\exp \int_{0}^{z}\frac{h_{k,\gamma }(w(\xi ))-1}{\zeta }\,d\xi , $$

where $w(z)$ is analytic in E with $w(0)=0$ and $\vert w(z) \vert <1$. Moreover, for $\vert z \vert =\rho$, we have

$$\begin{aligned} \exp \biggl( \int_{0}^{1}\frac{h_{k,\gamma }(-\rho )-1}{\rho }\,d\rho \biggr)& \leq \biggl\vert \frac{S_{q}^{m}f(z)}{z} \biggr\vert \\ &\leq \exp \biggl( \int_{0}^{1}\frac{h_{k,\gamma }(\rho )-1}{\rho }\,d\rho \biggr) , \end{aligned}$$

where $h_{k,\gamma }(z)$ is defined by (1.6).

Theorem 3.3

If $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$, then

$$ \vert a_{p+1} \vert \leq \frac{\delta }{ \{ [p+1]_{q}-p \} \psi _{p+1}} $$

(3.6)

and

$$ \vert a_{n+p-1} \vert \leq \frac{\delta }{ \{ [n+p-1]_{q}-p \} \psi _{n+p-1}}\prod _{j=1}^{n-2} \biggl( 1+\frac{\delta }{ \{ [j+p]_{q}-p \} } \biggr) \quad \textit{for }n=3,4,\ldots, $$

(3.7)

where $\delta =p \vert Q_{1} \vert$ with $Q_{1}$ given by (2.2).

Proof

Let

$$ \begin{aligned} &\frac{1}{[p]_{q}} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) =h(z), \\ &z\partial _{q}S_{q,p}^{m}f(z) = [p]_{q}S_{q,p}^{m}f(z)h(z), \end{aligned} $$

(3.8)

where $h(z)$ is analytic in E and $h(0)=1$. Let $h(z)=1+\sum _{n=1}^{\infty }c_{n}z^{n}$ and $S_{q,p}^{m}f(z)$ be given by (1.3). Then (3.8) becomes

$$ z^{p}+\sum_{n=p+1}^{\infty }[n]_{q} \psi _{n}a_{n}z^{n}=p \Biggl( \sum _{n=0}^{\infty }c_{n}z^{n} \Biggr) \Biggl( z^{p}+\sum_{n=p+1}^{\infty }\psi _{n}a_{n}z^{n} \Biggr) . $$

Now comparing the coefficients of $z^{n+p-1}$, we obtain

$$\begin{aligned} &[ n+p-1]_{q}\psi _{n+p-1}a_{n+p-1} =p\psi _{n+p-1}a_{n+p-1}+p \{ c_{1}\psi _{n+p-2}a_{n+p-2}+ \cdots+c_{n-1} \} , \\ &\bigl\{ {}[ n+p-1]_{q}-p \bigr\} \psi _{n+p-1}a_{n+p-1} =p \{ c_{1}\psi _{n+p-2}a_{n+p-2}+ \cdots+c_{n-1} \} . \end{aligned}$$

Taking the absolute on both sides and then applying the coefficient estimates $\vert c_{n} \vert \leq \vert Q_{1} \vert$, see in [23], we have

$$ \vert a_{n+p-1} \vert \leq \frac{p \vert Q_{1} \vert }{ \{ [n+p-1]_{q}-p \} \psi _{n+p-1}} \bigl\{ 1+\psi _{p+1} \vert a_{p+1} \vert +\cdots+\psi _{n+p-2} \vert a_{n+p-2} \vert \bigr\} . $$

Let us take $\delta =p \vert Q_{1} \vert$, then we have

$$ \vert a_{n+p-1} \vert \leq \frac{\delta }{ \{ [n+p-1]_{q}-p \} \psi _{n+p-1}} \bigl\{ 1+\psi _{p+1} \vert a_{p+1} \vert +\cdots+\psi _{n+p-2} \vert a_{n+p-2} \vert \bigr\} . $$

(3.9)

We apply mathematical induction on (3.9), so for $n=2$ in (3.9), we have

$$ \vert a_{p+1} \vert \leq \frac{\delta }{ \{ [p+1]_{q}-p \} \psi _{p+1}}, $$

(3.10)

which shows that (3.7) holds for $n=2$. Now consider the case $n=3$ in (3.9), we have

$$ \vert a_{p+2} \vert \leq \frac{\delta }{ \{ [p+2]_{q}-p \} \psi _{p+2}}\bigl\{ 1+\psi _{p+1} \vert a_{p+1} \vert \bigr\} . $$

Using (3.10), we have

$$ \vert a_{p+2} \vert \leq \frac{\delta }{ \{ [p+2]_{q}-p \} \psi _{p+2}}\biggl\{ 1+ \frac{\delta }{[p+1]_{q}-p}\biggr\} , $$

which shows that (3.7) holds for $n=3$. Let us assume that (3.7) is true for $n\leq t$, that is,

$$ \vert a_{t+p-1} \vert \leq \frac{\delta }{ \{ [t+p-1]_{q}-p \} \psi _{t+p-1}}\prod _{j=1}^{t-2} \biggl( 1+\frac{\delta }{[j+p]_{q}-p} \biggr) \quad \text{for }n=3,4,\ldots. $$

Consider

$$\begin{aligned} \vert a_{t+p} \vert \leq &\frac{\delta }{ \{ [t+p]_{q}-p \} \psi _{t+p}} \bigl\{ 1+\psi _{p+1} \vert a_{p+1} \vert +\cdots\psi _{t+p-1} \vert a_{t+p-1} \vert \bigr\} \\ \leq &\frac{\delta }{ \{ [t+p]_{q}-p \} \psi _{t+p}}\left \{ \textstyle\begin{array}{c} 1+\frac{\delta }{[p+1]_{q}-p}+\frac{\delta }{[p+2]_{q}-p} ( 1+\frac{\delta }{[p+1]_{q}-p} ) +\cdots \\ {}+\frac{\delta }{ \{ [t+p-1]_{q}-p \} }\prod_{j=1}^{t-2} ( 1+\frac{\delta }{[j+p]_{q}-p} ) \end{array}\displaystyle \right \} \\ =&\frac{\delta }{ \{ [t+p]_{q}-p \} \psi _{t+p}}\prod_{j=1}^{t-1} \biggl( 1+\frac{\delta }{[j+p]_{q}-p} \biggr) , \end{aligned}$$

which proves the assertion of theorem $n=t+1$. Hence (3.7) holds for all n, $n\geq 3$.

This completes the proof. □

When $p=1$, we have the following known result proved by Saqib et al. in [12].

Corollary 3.4

([12])

If $f(z)\in k-\mathcal{US}(q,\gamma ,m)$, then

$$ \vert a_{2} \vert \leq \frac{\delta }{ \{ [2]_{q}-1 \} [2]_{q}^{m}} $$

and

$$ \vert a_{n} \vert \leq \frac{\delta }{ \{ [n]_{q}-1 \} [n]_{q}^{m}}\prod _{j=1}^{n-2} \biggl( 1+\frac{\delta }{[j+1]_{q}-1} \biggr) \quad \textit{for }n=3,4,\ldots, $$

where $\delta = \vert Q_{1} \vert$ with $Q_{1}$ given by (2.2).

Theorem 3.5

Let $0\leq k<\infty$ be fixed and let $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$ with the form (1.1). Then, for a complex number μ,

$$ \bigl\vert a_{p+2}-\mu a_{p+1}^{2} \bigr\vert \leq \frac{pQ_{1}}{2 [ 2p+1 ] _{q}^{m} \{ [p+2]_{q}-p \} }\max \bigl[ 1, \vert 2v-1 \vert \bigr] , $$

(3.11)

where

$$ v=\frac{1}{2} \biggl\{ 1-\frac{Q_{2}}{Q_{1}}-Q_{1} \biggl( \frac{4p}{ \{ [p+1]_{q}-p \} }-\mu \frac{4p [ 2p+1 ] _{q}^{m} \{ [p+2]_{q}-p \} }{ ( [ 2p ] _{q}^{m} ) ^{2} \{ [p+1]_{q}-p \} } \biggr) \biggr\} , $$

(3.12)

and $\delta =p \vert Q_{1} \vert$, with $Q_{1}$ and $Q_{2}$ given by (2.2) and (2.3).

Proof

Let $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$, then there exists a Schwarz function $w(z)$, with $w(0)=0$ and $|w(z)|<1$, such that

$$\begin{aligned} \begin{aligned} &\frac{1}{[p]_{q}} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) \prec h_{k,\gamma }(z),\quad z\in E, \\ &\frac{1}{[p]_{q}} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) =h_{k,\gamma }\bigl(w(z)\bigr). \end{aligned} \end{aligned}$$

(3.13)

Let $h(z)\in \mathcal{P}$ be a function defined as

$$ h(z)=\frac{1+w(z)}{1-w(z)}, $$

which gives

$$ w(z)=\frac{c_{1}}{2}z+\frac{1}{2}\biggl(c_{2}- \frac{c_{1}^{2}}{2}\biggr)z^{2}+\cdots $$

and

$$ h_{k,\gamma }\bigl(w(z)\bigr)=1+\frac{Q_{1}c_{1}}{2}z+ \biggl\{ \frac{Q_{2}c_{1}^{2}}{4}+\frac{1}{2}\biggl(c_{2}- \frac{c_{1}^{2}}{2}\biggr)Q_{1} \biggr\} z^{2}+\cdots. $$

(3.14)

Using (3.14) in (3.13) and along with (1.3), we obtain

$$ a_{p+1}=\frac{pQ_{1}c_{1}}{ [ 2p ] _{q}^{m} \{ [p+1]_{q}-p \} } $$

and

$$ a_{p+2}=\frac{p}{ [ 2p+1 ] _{q}^{m} \{ [p+2]_{q}-p \} } \biggl\{ \frac{Q_{1}c_{2}}{2}+ \frac{c_{1}^{2}}{4} \biggl( Q_{2}-Q_{1}+\frac{4pQ_{1}^{2}}{ \{ [p+1]_{q}-p \} } \biggr) \biggr\} . $$

Using any complex number μ and the above coefficients, we have

$$ a_{p+2}-\mu a_{p+1}^{2}=\frac{pQ_{1}}{2 [ 2p+1 ] _{q}^{m} \{ [p+2]_{q}-p \} } \bigl\{ c_{2}-vc_{1}^{2} \bigr\} . $$

(3.15)

Using Lemma 2.3 on (3.15), we have

$$ \bigl\vert a_{p+2}-\mu a_{p+1}^{2} \bigr\vert \leq \frac{pQ_{1}}{2 [ 2p+1 ] _{q}^{m} \{ [p+2]_{q}-p \} }\max \bigl[ 1, \vert 2v-1 \vert \bigr] , $$

where

$$ v=\frac{1}{2} \biggl\{ 1-\frac{Q_{2}}{Q_{1}}-Q_{1} \biggl( \frac{4p}{ \{ [p+1]_{q}-p \} }-\mu \frac{4p [ 2p+1 ] _{q}^{m} \{ [p+2]_{q}-p \} }{ ( [ 2p ] _{q}^{m} ) ^{2} \{ [p+1]_{q}-p \} } \biggr) \biggr\} . $$

This is our required result (3.11). □

When $p=1$, we have the following known result proved by Saqib et al. in [12].

Corollary 3.6

([12])

Let $0\leq k<\infty$ be fixed and let $f(z)\in k-\mathcal{US}(q,\gamma ,m)$ with the form (1.1). Then, for a complex number μ,

$$ \bigl\vert a_{3}-\mu a_{2}^{2} \bigr\vert \leq \frac{Q_{1}}{2 [ 3 ] _{q}^{m} \{ [3]_{q}-1 \} }\max \bigl[ 1, \vert 2v-1 \vert \bigr] , $$

where

$$ v=\frac{1}{2} \biggl\{ 1-\frac{Q_{2}}{Q_{1}}-Q_{1} \biggl( \frac{4}{[2]_{q}-1}-\mu \frac{4 [ 3 ] _{q}^{m} \{ [3]_{q}-1 \} }{ ( [ 2 ] _{q}^{m} ) ^{2} \{ [2]_{q}-1 \} } \biggr) \biggr\} , $$

and $Q_{1}$ and $Q_{2}$ are given by (2.2) and (2.3).

Theorem 3.7

If a function $f(z)\in \mathcal{A}_{p}$ has the form (1.1) and satisfies the condition

$$ \sum_{n=p+1}^{\infty } \bigl\{ \bigl\{ \bigl\vert [n]_{q}-p \bigr\vert \bigr\} (k+1)+p \vert \gamma \vert \bigr\} \vert \psi _{n} \vert \vert a_{n} \vert \leq \vert \gamma \vert \vert p \vert , $$

(3.16)

then $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$.

Proof

Let

$$\begin{aligned} \biggl\vert \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\vert =& \biggl\vert \frac{z\partial _{q}S_{q,p}^{m}f(z)-pS_{q,p}^{m}f(z)}{pS_{q,p}^{m}f(z)} \biggr\vert \\ =& \biggl\vert \frac{\sum_{n=p+1}^{\infty }\psi _{n} \{ [n]_{q}-p \} a_{n}z^{n}}{pz^{p}+p\sum_{n=p+1}^{\infty }\psi _{n}a_{n}z^{n}} \biggr\vert \\ \leq &\frac{\sum_{n=p+1}^{\infty } \vert \psi _{n} \{ [n]_{q}-p \} \vert \vert a_{n} \vert }{ \vert p \vert -\sum_{n=p+1}^{\infty }p \vert \psi _{n} \vert \vert a_{n} \vert }. \end{aligned}$$

(3.17)

From (3.16), it follows that

$$ p-\sum_{n=p+1}^{\infty }p \vert \psi _{n} \vert \vert a_{n} \vert >0. $$

To show that $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$, it is enough to prove that

$$ \biggl\vert \frac{k}{\gamma } \biggl\{ \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\} \biggr\vert -\operatorname{Re} \biggl\{ \frac{1}{\gamma } \biggl\{ \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\} \biggr\} \leq 1. $$

From (3.17), we have

$$\begin{aligned} & \biggl\vert \frac{k}{\gamma } \biggl\{ \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\} \biggr\vert -\operatorname{Re} \biggl\{ \frac{1}{\gamma } \biggl\{ \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\} \biggr\} \\ &\quad \leq \frac{k}{ \vert \gamma \vert } \biggl\vert \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\vert +\frac{1}{ \vert \gamma \vert } \biggl\vert \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\vert \\ &\quad \leq \frac{(k+1)}{ \vert \gamma \vert } \biggl\vert \frac{1}{p} \biggl( \frac{z\partial _{q}S_{q,p}^{m}f(z)}{S_{q,p}^{m}f(z)} \biggr) -1 \biggr\vert \\ &\quad =\frac{(k+1)}{ \vert \gamma \vert } \biggl\vert \frac{z\partial _{q}S_{q,p}^{m}f(z)-pS_{q,p}^{m}f(z)}{pS_{q,p}^{m}f(z)} \biggr\vert \\ &\quad \leq \frac{(k+1)}{ \vert \gamma \vert } \biggl\{ \frac{\sum_{n=p+1}^{\infty } \vert \psi _{n} \{ [n]_{q}-p \} \vert \vert a_{n} \vert }{ \vert p \vert -\sum_{n=p+1}^{\infty }p \vert \psi _{n} \vert \vert a_{n} \vert } \biggr\} \\ &\quad \leq 1. \end{aligned}$$

 □

When $p=1$, we have the following known result proved by Hussain et al. in [12].

Corollary 3.8

([12])

If a function $f(z)\in \mathcal{A}$ has the form (1.1) and satisfies the condition

$$ \sum_{n=2}^{\infty } \bigl\{ \bigl\vert [n]_{q}-1 \bigr\vert (k+1)+ \vert \gamma \vert \bigr\} [n]_{q}^{m} \vert a_{n} \vert \leq \vert \gamma \vert , $$

then $f(z)\in k-\mathcal{US}(q,\gamma ,m)$.

When $q\rightarrow 1$, $p=1$, $m=0$, $\gamma =1-\alpha$, with $0\leq \alpha <1$, we have the following known result, proved by Shams et al. in [28].

Corollary 3.9

A function $f\in A$ of the form (1.1) is in the class $\mathcal{SD}(k,\alpha )$ if it satisfies the condition

$$ \sum_{n=2}^{\infty } \bigl\{ n(k+1)-(k+\alpha ) \bigr\} \vert a_{n} \vert \leq 1-\alpha , $$

where $0\leq \alpha <1$ and $k\geq 0$.

When $q\rightarrow 1$, $p=1$, $m=0$, $\gamma =1-\alpha$, with $0\leq \alpha <1$ and $k=0$, we have the following known result proved by Silverman in [30].

Corollary 3.10

A function $f\in A$ of the form (1.1) is in the class $\mathcal{SD}(\alpha )$ if it satisfies the condition

$$ \sum_{n=2}^{\infty } \{ n-\alpha \} \vert a_{n} \vert \leq 1-\alpha . $$

Theorem 3.11

Let $f(z)\in k-\mathcal{US}(q,\gamma ,m,p)$. Then $f(E)$ contains an open disk of radius

$$ r=\frac{ \{ [p+1]_{q}-p \} \psi _{p+1}}{(p+1) \{ [p+1]_{q}-p \} \psi _{p+1}+\delta }, $$

where $\delta =p \vert Q_{1} \vert$ with $Q_{1}$ given by (2.2).

Proof

Let $w_{0}\neq 0$ be a complex number such that $f(z)\neq w_{0}$ for $z\in E$. Then

$$ f_{1}(z)=\frac{w_{0}f(z)}{w_{0}-f(z)}=z+ \biggl( a_{p+1}+ \frac{1}{w_{0}} \biggr) z^{p+1}+\cdots. $$

Since $f_{1}(z)$ is univalent, so

$$ \biggl\vert a_{p+1}+\frac{1}{w_{0}} \biggr\vert \leq p+1. $$

Now, by using (3.6), we have

$$ \biggl\vert \frac{1}{w_{0}} \biggr\vert \leq \frac{(p+1) \{ [p+1]_{q}-p \} \psi _{p+1}+\delta }{ \{ [p+1]_{q}-p \} \psi _{p+1}}. $$

Hence we have

$$ \vert w_{0} \vert \geq \frac{ \{ {}[ p+1]_{q}-p \} \psi _{p+1}}{(p+1) \{ [p+1]_{q}-p \} \psi _{p+1}+\delta }. $$

 □

When $p=1$, we have the following known result proved by Saqib et al. in [12].

Corollary 3.12

([12])

Let $f(z)\in k-\mathcal{US}(q,\gamma ,m)$. Then $f(E)$ contains an open disk of radius

$$ r=\frac{ \{ [2]_{q}-1 \} [ 2 ] _{q}^{m}}{2 [ 2 ] _{q}^{m} \{ [2]_{q}-1 \} +Q_{1}}, $$

where $Q_{1}$ is given by (2.2).