Certain properties of multivalent analytic functions defined by $q$-difference operator involving the Janowski function

1.   Introduction

Quantum calculus is ordinary classical calculus without the concept of limits. In recent years, the area of $q$-calculus has attracted the serious attention of researchers. This great interest is due to its application in various branches of mathematics and physics. The application of $q$-calculus was initiated by Jackson ^[9,10]. He was the first to develop $q$-derivative and $q$-integral in a systematic way. Later, geometrical interpretation of $q$-analysis has been recognized through studies on quantum groups. It also suggests a relation between integrable systems and $q$-analysis. Aral ^[5] and Anastassiou and Gal ^[2,3] generalized some complex operators which are known as $q$-Picard and $q$-Gauss-Weierstrass singular integral operators. Moreover, Srivastava et al. published a set of articles ^{[13,15,16,17,18,19]} in which they concentrated upon the classes of $q$-starlike functions related with the Janowski functions ^[11] from several different aspects. Additionally, a recently-published survey-cum-expository review article by Srivastava ^[20] is potentially useful for researchers and scholars working on these topics. In this survey-cum-expository review article ^[20], the mathematical explanation and applications of the fractional $q$-calculus and the fractional $q$-derivative operators in Geometric Function Theory was systematically investigated. Refer to further $q$-theory can be found in ^{[1,6,7,8,12,14]}.

Let $A_p$ denote the class of multivalent analytic functions $f(z)$ given by Taylor-Maclaurin's series

in the open unit disk $D = \{z:|z| < 1\}$. For $p = 1$, we write $A: = A_1$.

A function $f(z)\in A_p$ is said to be multivalent starlike function of order $\sigma$ and is written as $f(z)\in S^*_p(\sigma)$, if it satisfies

for all $z\in D$ (see ^[4]).

A function $f(z)\in A_p$ is known as multivalent convex function of order $\sigma$ and is denoted by $f(z)\in C_p(\sigma)$, if it satisfies

for all $z\in D$.

Given two functions $f(z)$ and $g(z)$, which are analytic in $D$, we say that the function $g(z)$ is subordinate to $f(z)$ and write $g(z)\prec f(z)$ $(z\in D)$, if there exists a Schwarz function $w(z)$ such that $g(z) = f(w(z))$ $(z\in D)$. In particular, if $f(z)$ is univalent in $D$, then we have the following equivalence:

Definition 1. A function $h(z)$ is said to be in the class $P[A, B]$, if it is analytic in $D$ with $h(0) = 1$ and

equivalently we can write

Definition 2. Let $q\in (0, 1)$ and define the $q$-number $[\lambda]_q$ by

Particularly, when $\lambda = 0$, we have $[0]_q = 0$.

Definition 3 ^[9,10]. Let $q\in (0, 1)$. The $q$-difference operator $\partial_q$ of a function $f(z)$ is defined by

provided that $f'(0)$ exists.

From Definition 3, we can see that

for a differentiable function $f(z)$ in a given subset of $C$. Also, for $f(z) = z^p+\sum^\infty_{n = 1}a_{p+n}z^{p+n}$, one can observe that (see ^[21])

where $[p]_q = \frac{1-q^p}{1-q} = 1+q+q^2+\cdots+q^{p-1}$.

In $q$-calculus sense, we now define the following subclass of $A_p$ associated with the $q$-difference operator $\partial_q$.

Definition 4. A function $f(z)\in A_p$ $(p\geq 2)$ is said to belong to the class $T_{p, q}(\alpha, A, B)$, if it satisfies

or equivalently

Remark. For $\alpha = 0$, $A = 1$, $B = -1$ and $q\rightarrow1^{-}$, we can see from Definition 4 that $T_{p, q}(\alpha, A, B)$ reduces to the subclass of $p$-valently close-to-convex functions.

In this paper we shall study some geometric properties of functions in $T_{p, q}(\alpha, A, B)$ such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radii of starlikeness and convexity, partial sums and closure theorems.

2.   Main results

Theorem 1. Let $p\geq 2$ and $\frac{[p-1]_q}{[p]_q}\leq\alpha < 1$. Also let $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in A_p$. Then $f(z)\in T_{p, q}(\alpha, A, B)$ if and only if

Proof. Assuming that the inequality (2.1) holds true, then we only need to show the inequality (1.1). Now we have

which shows that $f(z)\in T_{p, q}(\alpha, A, B)$.

Conversely, let $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$. Then from (1.1), we have

The inequality (2.2) is true for all $z\in D$. Now we choose $z = \mathrm{Re}z\rightarrow1^{-}$ and obtain the inequality (2.1). Thus the proof of Theorem 1 is completed

Corollary 1. Let $\frac{[p-1]_q}{[p]_q} < \alpha < 1$ and $-1 < B < A\leq 1$. If $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$, then

The results are sharp for the function $f(z)$ defined by

Theorem 2. Let $\frac{[p-1]_q}{[p]_q} < \alpha < 1$ and $-1 < B < A\leq 1$. If $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$, then for $|z| = r$, we have

where

The bounds are sharp for the function

Proof. Let $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}$. By applying the triangle inequality, we have

Since $|z| = r < 1$, we can see that $r^{p+n}\leq r^{p+1}$. Thus we have

and

Considering $f(z)\in T_{p, q}(\alpha, A, B)$, we know from Theorem 1 that

As we know that $\{(1+B)[p+n]_q\left(\alpha[p+n-1]_q-[p-1]_q\right)\}$ is an increasing sequence with respect to $n$ $(n\geq1)$, so

Hence by transitivity we obtain

which implies that

Substituting inequality (2.5) into inequalities (2.3) and (2.4), we get the required results. The proof of Theorem 2 is completed.

Theorem 3. Let $\frac{[p-1]_q}{[p]_q} < \alpha < 1$ and $-1 < B < A\leq 1$. If $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$, then for $|z| = r$, we have

where

The results are sharp for the function

Proof. Let $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}$. From Definition 3, we can write

By applying the triangle inequality, we get

Further, we have that

and

Since $f(z)\in T_{p, q}(\alpha, A, B)$, we know from Theorem 1 that

As we know that $\{(1+B)\left(\alpha[p+n-1]_q-[p-1]_q\right)\}$ is an increasing sequence in regard to $n$ $(n\geq 1)$, so

Thus by transitivity we have

which implies that

By putting (2.8) in (2.6) and (2.7), we obtain the required results. Now Theorem 3 is proved.

Theorem 4. Let $\frac{[p-1]_q}{[p]_q} < \alpha < 1$, $-1 < B < A\leq 1$ and $0\leq\sigma < p$. If

then for $0 < |z| < r_1$, $f(z)$ is $p$-valently starlike function of order $\sigma$, where

Proof. Let $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$. In order to prove $f(z)\in S_p^*(\sigma)$, we only need to show

The subordination above is equivalent to $\left|\frac{zf'(z)-pf(z)}{zf'(z)+(p-2\sigma)f(z)}\right| < 1$. After some calculations and simplifications, we obtain

From the inequality (2.1), we can obviously know that

Inequality (2.9) wants to be true if it satisfies the following inequality

which implies that

or

Let $r_1 = \min\left\{\inf_{n\geq 1}\left(\frac{(p-\sigma)(1+B)[p+n]_q(\alpha[p+n-1]_q-[p-1]_q)}{(n+p-\sigma)(1-\alpha)(A-B)[p]_q[p-1]_q}\right)^{\frac{1}{n}}, 1\right\}$, then we get the required result. The proof of Theorem 4 is completed.

Theorem 5. Let $\frac{[p-1]_q}{[p]_q} < \alpha < 1$, $-1 < B < A\leq 1$ and $0\leq\sigma < p$. If

then for $0 < |z| < r_2$, $f(z)$ is $p$-valently convex function of order $\sigma$, where

Proof. Let $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$. To prove $f(z)\in C_p(\sigma)$, we must show that

This is equivalent to the inequality $\left|\frac{zf''(z)-(p-1)f'(z)}{zf''(z)+(1-2\sigma+p)f'(z)}\right| < 1$. After some calculations and simplifications, we have

From the inequality (2.1), we can easily obtain that

For inequality (2.10) to be true, it will be enough if

which implies that

or

Let $r_2 = \min\left\{\inf_{n\geq 1}\left(\frac{p(p-\sigma)(1+B)[p+n]_q(\alpha[p+n-1]_q-[p-1]_q)}{(p+n)(n+p-\sigma)(1-\alpha)(A-B)[p]_q[p-1]_q}\right)^{\frac{1}{n}}, 1\right\}$, then we obtain the required result. Now Theorem 5 is proved.

Next, we will study the ratio of a function $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}$ to its sequence of partial sums $f_k(z) = z^p-\sum^k_{n = 1}|a_{p+n}|z^{p+n}(k = 1, 2, 3, \cdots)$ for all $z\in D$.

Theorem 6. Let $\frac{(1+B)[p+1]_q[p-1]_q+(A-B)[p]_q[p-1]_q}{(1+B)[p+1]_q[p]_q+(A-B)[p]_q[p-1]_q} < \alpha < 1$, $-1 < B < A\leq 1$. If $f(z) = z^p-\sum^\infty_{n = 1}|a_{p+n}|z^{p+n}\in T_{p, q}(\alpha, A, B)$, then we have

and

where

Proof. In order to prove (2.11), we set

After some simplifications, we have

and

Now we can see that $|w(z)| < 1$, if and only if

From (2.1) we have $\sum^\infty_{n = 1}\varphi_n|a_{p+n}|\leq 1$. It is not difficult to see that $\varphi_n$ is an increasing sequence with respect to $n$ and that $\varphi_n\geq 1$ $(n = 1, 2, \cdots)$. Therefore, we get

Thus, the inequality (2.14) is true. This proves (2.11).

Next, in order to prove the inequality (2.12), we consider

After some simplifications, we can find that

and

Now we can see that $|w(z)| < 1$ if it satisfies

The remaining part of the proof is much akin to that of (2.11) and hence we omit it. The proof of the Theorem is completed.

Theorem 7. Let $\frac{[p-1]_q}{[p]_q}\leq\alpha < 1$. If $f_j(z) = z^p-\sum^\infty_{n = 1}|a_{p+n, j}|z^{p+n}\in T_{p, q}(\alpha, A, B)$ $(j = 1, 2)$, then for $0\leq\lambda\leq 1$, the function $H(z) = \lambda f_1(z)+(1-\lambda)f_2(z)\in T_{p, q}(\alpha, A, B)$.

Proof. For $0\leq\lambda\leq 1$, the function $H(z)$ can be written as

For functions $f_1(z), f_2(z)\in T_{p, q}(\alpha, A, B)$, by Theorem 1, we have

which shows that $H(z)\in T_{p, q}(\alpha, A, B)$.

Corollary 2. Let $\frac{[p-1]_q}{[p]_q}\leq\alpha < 1$. If $f_j(z) = z^p-\sum^\infty_{n = 1}|a_{p+n, j}|z^{p+n}\in T_{p, q}(\alpha, A, B)$ $(j = 1, 2, \cdots, t)$, then the function $F(z) = \sum^t_{j = 1}\lambda_jf_j(z)\in T_{p, q}(\alpha, A, B)$, where $\lambda_j\geq 0$ and $\sum^t_{j = 1}\lambda_j = 1$.

Theorem 8. Let $\frac{[p-1]_q}{[p]_q}\leq\alpha < 1$. If $f_j(z) = z^p-\sum^\infty_{n = 1}|a_{p+n, j}|z^{p+n}\in T_{p, q}(\alpha, A, B)$ $(j = 1, 2)$, then for $-1\leq m\leq 1$, we have

Proof. For $-1\leq m\leq 1$, the function $Q_m(z)$ can be written as

In view of $f_1(z), f_2(z)\in T_{p, q}(\alpha, A, B)$, by applying Theorem 1, we get

which shows that $Q_m(z)\in T_{p, q}(\alpha, A, B)$.

3.   Conclusions

Our objective is to generalize some classical interesting results in geometric function theory from the ordinary analysis to $q$-analysis. By using the $q$-difference operator, a new subclass $T_{p, q}(\alpha, A, B)$ of multivalent analytic functions is introduced. Some geometric properties of functions in $T_{p, q}(\alpha, A, B)$ such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radii of starlikeness and convexity, partial sums and closure theorems are given. In particular, if we let $\alpha = 0$, $A = 1$, $B = -1$ and $q\rightarrow1^{-}$, then $T_{p, q}(\alpha, A, B)$ reduces to the subclass of $p$-valently close-to-convex functions.

Acknowledgments

The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped us to improve the paper. This work was supported by National Natural Science Foundation of China (Grant No.11571299).

Conflict of interest

The authors declare no conflict of interest.