Meromorphic Parabolic Starlike Functions Associated with 𝑞-Hypergeometric Series

Abstract

We introduce a new class of meromorphic parabolic starlike functions with a fixed point defined in the punctured unit disk involving the -hypergeometric functions. We obtained coefficient inequalities, growth and distortion inequalities, and closure results for functions . We further established some results concerning convolution and the partial sums.

1. Introduction

Let be a fixed point in the unit disc . Denote by the class of functions which are regular and Also denote by , the subclass of consisting of the functions of the form which are analytic in . Note that is subclasses of consisting of univalent functions in . By and , respectively, we mean the classes of analytic functions that satisfy the analytic conditions , and , for introduced and studied by Kanas and Ronning [1]. The class is defined by geometric property that the image of any circular arc centered at is starlike with respect to and the corresponding class is defined by the property that the image of any circular arc centered at is convex. We observe that the definitions are somewhat similar to the ones introduced by Goodman in [2, 3] for uniformly starlike and convex functions, except that in this case the point is fixed. In particular, and , respectively, are the well-known standard classes of convex and starlike functions.

Let denote the class of meromorphic functions of the form defined on the punctured unit disk .

Denote by the subclass of consisting of the functions of the form

A function of the form (4) is in the class of meromorphic starlike of order () denoted by , if and is in the class of meromorphic convex of order () denoted by , if

For functions given by (4) and ,   we define the Hadamard product or convolution of and by

More recently, Purohit and Raina [4] introduced a generalized -Taylor’s formula in fractional -calculus and derived certain -generating functions for -hypergeometric functions. In this work we proceed to derive a generalized differential operator on meromorphic functions in involving these functions and discuss some of their properties.

For complex parameters and   the -hypergeometric function is defined by with where when .

The -shifted factorial is defined for as a product of factors by and in terms of basic analogue of the gamma function It is of interest to note that is the familiar Pochhammer symbol and Now for , , and , the basic hypergeometric function defined in (8) takes the form which converges absolutely in the open unit disk .

Corresponding to the function recently for meromorphic functions consisting functions of the form (3), Huda and Darus [5] introduce -analogue of Liu-Srivastava operator as below: where .

In this paper for functions and for real parameters and   we define the following new linear operator: as where

Throughout our study for , we let unless otherwise stated.

Motivated by earlier works on meromorphic functions by function theorists (see [6–14]), we define the following new subclass of functions in by making use of the generalized operator .

For and , we let denote a subclass of consisting functions of the form (4) satisfying the condition that where is given by (17).

Further, shortly we can state this condition by where

It is of interest to note that, on specializing the parameters , and , , we can define various new subclasses of . We illustrate two important subclasses in the following examples.

Example 1. For , we let denote a subclass of consisting functions of the form (4) satisfying the condition that where is given by (17).

Example 2. For , we let denote a subclass of consisting functions of the form (4) satisfying the condition that where is given by (17).

In this paper, we obtain the coefficient inequalities, growth and distortion inequalities, and closure results for the function class . Properties of certain integral operator and convolution properties of the new class are also discussed.

2. Coefficients Inequalities

In order to obtain the necessary and sufficient condition for a function, , we recall the following lemmas.

Lemma 3. If is a real number and is a complex number, then .

Lemma 4. If is a complex number and , are real numbers, then

Analogous to the lemma proved by Dziok et al. [8], we state the following lemma without proof.

Lemma 5. Suppose that , , and the function is of the form , , with , then

Theorem 6. Let be given by (4). Then if and only if

Proof. If , then by (20) we have Making use of Lemma 4, where is given by (21). Substituting , and letting , we have This shows that (26) holds.
Conversely, assume that (26) holds. Since , if and only if , it is sufficient to show that Using (26) and taking , we get Thus we have .

For the sake of brevity throughout this paper we let unless otherwise stated.

Our next result gives the coefficient estimates for functions in .

Theorem 7. If , then The result is sharp for the functions given by

Proof. If , then we have, for each , Therefore we have Since satisfies the conditions of Theorem 6, and the equality is attained for this function.

Theorem 8. Suppose that there exists a positive number : If , then If , then the result is sharp for

Proof. Let and be given by (4) Since , and by Theorem 6, Using this, we have Similarly The result is sharp for function (40) with Similarly we can prove the other inequality .

3. Order of Starlikeness

In the following theorem we obtain the order of starlikeness for the class . We say that given by (4) is meromorphically starlike of order , , in when it satisfies condition (5) in .

Theorem 9. Let the function given by (4) be in the class . Then, if there exists and it is positive, then is meromorphically starlike of order in .

Proof. Let the function be of the form (4). If , then by (46) for all . From (47) we get for all , and thus because of (26). Hence, from (49) and (25), is meromorphically starlike of order in .

Suppose that there exists a number , , such that each is meromorphically starlike of order in . The function is in the class ; thus it should satisfy (25) with while the left–hand side of (51) becomes which contradicts (51). Therefore the number in Theorem 9 cannot be replaced with a greater number. This means that is called radius of meromorphically starlikeness of order for the class .

4. Results Involving Modified Hadamard Products

For functions we define the Hadamard product or convolution of and by Let

Theorem 10. For functions defined by (53), let and . Then where and . The results are the best possible for where .

Proof. In view of Theorem 6, it suffices to prove that where is defined by (56) under the hypothesis. It follows from (26) and the Cauchy-Schwarz inequality that Thus we need to find the largest such that
By virtue of (59) it is sufficient to find the largest , such that which yields for where is given by (55) and, since is a decreasing function of , we have and , which completes the proof.

Theorem 11. Let the functions , , defined by (53) be in the class . Then where with .

Proof. By taking in the above theorem, the results follow.

For functions in the class , we can prove the following inclusion property.

Theorem 12. Let the functions defined by (53) be in the class . Then the function , defined by is in the class where and .

Proof. In view of Theorem 6, it is sufficient to prove that where ; we find from (53) and Theorem 6 that which would yield On comparing (67) and (69) it can be seen that inequality (66) will be satisfied if That is, if where is given by (55) and is a decreasing function of , we get (66), which completes the proof.

5. Closure Theorems

We state the following closure theorems for without proof (see [8–10]).

Theorem 13. Let the function be in the class for every . Then the function defined by belongs to the class , where , .

Theorem 14. Let and for . Then if and only if can be expressed in the form where and .

Theorem 15. The class is closed under convex linear combination.

Now, we prove that the class is closed under integral transforms.

Theorem 16. Let the function given by (4) be in . Then the integral operator is in , where
The result is sharp for the function .

Proof. Let . Then It is sufficient to show that Since , we have Note that (76) is satisfied if From (78), we have A simple computation will show that is increasing and . Using this, the results follow.

6. Partial Sums

Silverman [15] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. As a natural extension, one is interested in searching results analogous to those of Silverman for meromorphic univalent functions. In this section, motivated essentially by the work of Silverman [15] and Cho and Owa [16], we will investigate the ratio of a function of the form (4) to its sequence of partial sums. Consider when the coefficients are sufficiently small to satisfy the condition analogous to More precisely we will determine sharp lower bounds for and . In this connection we make use of the well-known results that , , if and only if satisfies the inequality .

Unless otherwise stated, we will assume that is of the form (4) and its sequence of partial sums is denoted by (80).

Theorem 17. Let be given by (4) which satisfies condition (26) and suppose that all of its partial sums (80) do not vanish in . Moreover, suppose that Then, where The result (83) is sharp with the function given by

Proof. Define the function by It suffices to show that ; hence we find that From condition (26), it readily yields the assertion (83) of Theorem 17.
To see that the function given by (85) gives the sharp result, we observe that for when which shows that the bound (83) is the best possible for each .

We next determine bounds for .

Theorem 18. Under the assumptions of Theorem 17, we have The result (89) is sharp with the function given by (85).

Proof. By setting and proceeding as in Theorem 17, we get the desired result and so we omit the details.

Concluding Remark. We observe that, if we specialize the parameters and as mentioned in Examples 1 and 2, we obtain the analogous results for the classes and . Further specializing the parameters , various other interesting results (as in Theorems 6–18) can be derived easily for the function class based on interesting differential operators as illustrated below.

(1) For , , , , , , the operator defined by Liu and Srivastava [10].

(2) For , , , , the operator was introduced and studied by Liu and Srivastava [9].

(3) For , , , , the operator , where is the differential operator which was introduced by Ganigi and Uralegaddi [17].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the referee for their valuable suggestions.