Research Article | Open Access
On a New Class of -Valent Meromorphic Functions Defined in Conic Domains
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.
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  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  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  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 ). Let be convex in and , where , , and . If is analytic in , with , then
Lemma 2 (see ). Let and and . If is univalent and convex in , then for .
Lemma 3 (see ). 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 haveSince , 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 havewhich 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.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work here is supported by AP-2013-009.
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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.