Some Properties of Certain Subclass of Meromorphically Multivalent Functions Defined By Linear Operator

Problem statement: By means of the Hadamard product (or convolution), a linear operator was introduced. This operator was motivated by many authors namely Srivastava, Swaminathan, Owa and many others. The operator was indeed needed to create new ideas in the area of geometric function theory. Approach: The linear operator of meromorphic p-valent functio ns was proposed and defined. The preliminary concept of subordination was introd uced to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. Results: Having the operator, subordination theorems established by using standard concept of s ubordination and reduced to well-known results studied by various authors. The operator was then a pplied for fractional calculus and obtained new subordination theorem. Conclusion: Therefore, interesting operators could be obtained with some earlier results and standard methods.

Also for a function f(z)∈∑ p , we define the integral operator J ν,p by: There are many researches (Uralegaddi and Somanatha, 1991;Whittaker and Watson, 1927;Yang, 1995;1996) in which the operator J ν,p was investigated.
which is increasing function of ℜ{λ} and 1 1 2 The estimate is sharp in the sense that the bound cannot be improved (Ponnusamy, 1995).
For real or complex numbers a, b,c, c z , We note that the above series converges absolutely for z∈U and hence represents an analytic function in the unit disc U (Whittaker and Watson, 1927).
Each of the identities (asserted by Lemma C) is fairly well known (Liu and Srivastava, 2004b;Whittaker and Watson, 1927).
The result follows by using the identity: In the following theorem, we shall extend the above results as follows.
Theorem 3: Suppose that the functions f(z) and g(z) are in ∑ p and suppose g(z) satisfies the condition (10). If: and Proof: Let: Then q(z) is analytic in U with q(0) = 1. Putting: We observe that by hypothesis, { } A simple computation shows that: I L 1, g(z) I L , f (z) q(z),zq (z) I L , g(z) Using the hypothesis (28), we obtain: This shows that ℜ{Ψ(ir 2 , s 1 )}∉Ω for each z∈U. Hence by Lemma A, we have ℜ{q(z)} > 0, z∈U. This proves (32). The proof of (33) follows by using (32) and (33) in the identity: I L 1, f (z) I L , f (z) I L 1, g(z) I L , g(z) I L , f (z) This completes the proof of Theorem 3.

DISCUSSION
By putting some values in Corollaries 2 and 3 and Theorem 3 we can obtain the following: • Putting λ = 1 and α, β > 0 in Corollary 2, we have (

CONCLUSION
The operator defined was motivated by various work studied earlier by the authors (Joshi and Srivastava, 1999;Liu and Srivastava, 2001;2004a;2004b). This operator can be generalized further and many other results such as the coefficient estimates and distortion theorem can be obtained.