On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

We introduce and study a subclass ΣP γ, k, λ, c of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ΣP γ, k, λ, c by fixing the second coefficient. Further, it is shown that the class ΣP γ, k, λ is closed under convex linear combination.


Introduction
Let Σ denote the class of functions of the form a n z n , 1.1 which are analytic in the punctured unit disk Let Σ S , Σ * γ and Σ K γ , 0 ≤ γ < 1 denote the subclasses of Σ that are meromorphically univalent, meromorphically starlike functions of order γ, and meromorphically convex 2 ISRN Mathematical Analysis functions of order γ, respectively.Analytically, f ∈ Σ * γ if and only if, f is of the form 1.1 and satisfies similarly, f ∈ Σ K γ , if and only if, f is of the form 1.1 and satisfies Let f, g ∈ Σ, where f z is given by 1.1 and g z is defined by Then the Hadamard product or convolution f * g of the functions f z and g z is defined by Let Σ P be the class of functions of the form that are analytic and univalent in U * .For complex parameters α 1 , . . ., α l and β 1 , . . ., β m β j / 0, −1, . . .; j 1, 2, . . ., m the generalized hypergeometric function l F m z is defined by where N denotes the set of all positive integers and θ n is the Pochhammer symbol defined by 1.9 12 unless otherwise stated in the sequel.We note that the linear operator H l m α 1 was earlier defined for multivalent functions by Dziok and Srivastava 8 and was investigated by Liu and Srivastava 7 .
Making use of the operator H l m α 1 , now we consider a subclass of functions in Σ P as follows.
Definition 1.1.For 0 ≤ γ < 1 and k ≥ 0 and 0 ≤ λ < 1/2, let Σ γ, k, λ denote a subclass of Σ consisting functions of the form 1.1 satisfying the condition that where H α 1 f z is given by 1.11 .Furthermore, we say that a function f ∈ Σ P γ, k, λ , whenever f z is of the form 1.7 .We observe that, by specializing the parameters l, m, α 1 , . . ., α l , β 1 , . . ., β m , k, γ and λ the class leads to various subclasses.As for illustrations, we present some examples for the cases.
Example 1.2.If l 2 and m 1 with α 1 1, α 2 1, β 1 1 and f z of the form 1.7 , then we obtain the new subclass M P γ, k, λ defined by , and f z of the form 1.7 , then we get the new subclass D δ P γ, k, λ defined by where is the differential operator which was introduced by Ganigi and Uralegaddi 9 .Also we note that the class D δ P γ, k, λ was introduced by Atshan and Kulkarni 10 .
, and f z of the form 1.7 , then we obtain the new subclass L P γ, k, λ defined by where the operator L α 1 ; β 1 f z was introduced and studied by Liu and Srivastava 11 see also 12, 13 .For the class Σ P γ, k, λ , the following characterization was given by Magesh et al. 14 .
For a function defined by 1.7 and in the class Σ P γ, k, λ , Theorem 1.5, immediately yields Hence we may take Motivated by Aouf and Darwish 1 , Aouf and Joshi 2 , Ghanim and Darus 12 , and Uralegaddi 15 , we now introduce the following class of functions and use the similar techniques to prove our results.
Let Σ P γ, k, λ, c be the subclass of Σ P γ, k, λ consisting functions in of the form In this paper, coefficient estimates, extreme points, growth and distortion bounds, radii of meromorphically starlikeness and convexity are discussed for the class Σ P γ, k, λ by fixing the second coefficient.Further, it is shown that the class Σ P γ, k, λ is closed under convex linear combination.

Main Results
In our first theorem, we now find out the coefficient inequality for the class Σ P γ, k, λ, c .
The result is sharp.
Proof.By putting in 1.17 , the result is easily derived.The result is sharp for the function Corollary 2.2.If the function f defined by 1.20 is in the class Σ P γ, k, λ, c , then The result is sharp for the function f z given by 2.3 .
Next we prove the following growth and distortion properties for the class Σ P γ, k, λ, c .

6 ISRN Mathematical Analysis
The result is sharp for the function f z given by Proof.Since Σ P γ, k, λ, c , Theorem 2.1 yields Thus, for 0 < |z| r < 1,

2.8
Thus the proof of the theorem is complete.

2.9
The result is sharp for the function f z given by

2.10
Proof.In view of Theorem 2.1, it follows that Thus, for 0 < |z| r < 1 and making use of 2.11 , we obtain

2.12
Hence the result follows.
Next, we will show that the class Σ P γ, k, λ, c is closed under convex linear combination.

2.14
Then f ∈ Σ P γ, k, λ, c if and only if it can expressed in the form where μ n ≥ 0 and ∞ n 1 μ n ≤ 1.

2.19
it follows that This completes the proof of the theorem.
Proof.Suppose that the function f be given by 1.20 , and let the function g be given by

2.21
Assuming that f and g are in the class Σ P γ, k, λ, c , it is enough to prove that the function h defined by is also in the class Σ P γ, k, λ, c .Since with the aid of Theorem 2.1.Thus, h ∈ Σ P γ, k, λ, c .
Next we determine the radii of meromorphically starlikeness and convexity of order δ for functions in the class Σ P γ, k, λ, c .Theorem 2.7.Let the function f z defined by 1.20 be in the class Σ P γ, k, λ, c , then i f is meromorphically starlike of order δ 0 ≤ δ < 1 in the disk |z| < r 1 γ, k, λ, c, δ where r 1 γ, k, λ, c, δ , is the largest value for which ii f is meromorphically convex of order δ 0 ≤ δ < 1 in the disk |z| < r 2 γ, k, λ, c, δ where r 2 γ, k, λ, c, δ , is the largest value for which

2.26
Each of these results is sharp for function f z given by 2.3 .

2.27
Thus, we have where Q denotes 2k γ 1 ,V denotes 1 − γ , and E denotes 1 − 2λ .Hence 2.28 holds true if and it follows that, from 2.1 , we may take where μ n ≥ 0 and ∞ n 2 μ n ≤ 1.For each fixed r, we choose the positive integer n 0 n 0 r for which

2.34
We find the value r 0 r 0 k, c, δ, n and the corresponding integer n 0 r 0 so that

2.35
It is the value for which the function f z is starlike in 0 < |z| < r 0 .ii In a similar manner, we can prove our result providing the radius of meromorphic convexity of order δ 0 ≤ δ < 1 for functions in the class Σ P γ, k, λ, c , so we skip the details of the proof of ii .