The Scientific World Journal
Volume 2016 (2016), Article ID 6360250, 7 pages
http://dx.doi.org/10.1155/2016/6360250

Research Article

On a New Class of -Valent Meromorphic Functions Defined in Conic Domains

Mohammed Ali Alamri and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 26 April 2016; Accepted 27 June 2016

Academic Editor: Jin-Lin Liu

Copyright © 2016 Mohammed Ali Alamri and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived.

1. Introduction

Let denote the class of functions of the form

which are analytic and -valent in the punctured unit disc centred at origin . Also by we mean is subordinate to which implies the existence of an analytic function, called Schwartz function with , for such that , where and are multivalent meromorphic functions. Note that if is univalent in then the above subordination is equivalent to and .

The set of points, for and ,

where

[1] showed that the extremal functions for conic regions are convex univalent and given by

where is Legendre’s complete elliptic integral of the first kind with as its complementary integral, , , and is chosen in such a way that .

The generalised hypergeometric function for complex parameters and with , for , is defined as

with , , and is the well-known Pochhammer symbol related to the factorial and the Gamma function by the relation

Also (5) implies

Liu and Srivastava [2] defined a linear operator for functions belonging to the class of multivalent meromorphic function as follows:

If we assume for brevity that then the following identity holds for this operator:

Shareef [3] defined and studied subclass of meromorphic function associated with conic domain, for , , and , as follows:

We now define a new subclass of meromorphic function associated with conic domain, for , , , and , as follows:

Since is a convex and univalent function, for it means is contained in , where

In the next two sections, for brevity, we drop the subscripts of the operator .

2. Preliminary Results

Lemma 1 (see [4]). Let be convex in and , where , , and . If is analytic in , with , then

Lemma 2 (see [5]). Let and and . If is univalent and convex in , then

for .

Lemma 3 (see [6]). If then

One now states and proves the main results.

3. Main Results

In this section we explore some of the geometric properties exhibited by the class .

We begin by discussing an inclusion property for the class .

Theorem 4. If then

Proof. Let and set

But differentiating (9) with respect to we get

Putting (18) in (17) we have

Taking logarithmic derivative of (19) we have

Since , therefore

Using Lemma 2 we have

provided or equivalently . Hence, .

We now show that the class is closed under a certain integral.

Theorem 5. If , then the integral

maps into .

Proof. From (23) we have

Note that

Differentiating (24) above we get

Differentiate again

Now let

Using (26) and (27) in (28) we get

Now taking logarithmic derivative we have

Using Lemma 2 we get which implies

This proves the assertion.

Now we get coefficient estimates of the class .

Theorem 6. If and is given by (1) then

provided

for all .

Proof. Let ; then, by definition, we have

which gives

Let us write ; then

Assuming , then (35) becomes

From (37) we have

Now comparing coefficients of we have

and comparing the coefficients of gives

and for the coefficient of we have

which generalise to

The above expression can also be written as

Now taking

we have

for all . Since is univalent and is convex, applying Rogosinski’s theorem we have

where is given in (15). Under the conditions given in (45), expressions (39)–(42) give

This can also be written as

This concludes the proof.

Competing Interests

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

Acknowledgments

The work here is supported by AP-2013-009.

References

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