Some Subordination Results Associated With Certain Subclass of Analytic Meromorphic Functions

For functions belonging to each of the subclasses Sw (β) and Cw (β) of normalized analytic functions in the open unit disk D, which are investigated in this paper when 0≤β<1, the authors derive several subordination results involving the Hadamard product (or convolution) of the associated functions. A number of interesting consequences of some of these subordination results are also discussed.


INTRODUCTION
Let w be a fixed point in D and A (w) = {f∈H (D): f(w) = f'(w)-1=0}. In [15] , Kanas and Ronning introduced the following classes S w = {f∈A (w): f is univalent in D} (z w)f (z) * C f A(w):Re 1 0,z D . w f (z) Late r Acu and Owa [1] studied the classes extensively. Let S w denoted the subclass of A(w) consisting of the function of the form: where α = Res (z,w), 0<α≤1 with z ≠ w. The class s * w is defined by geometric property that the image of any circular arc centered at w is starlike with respect to f (w) and the corresponding class C * w is defined by the property that the image of any circular arc centered at w is convex.
We observe that the definitions are somewhat similar to the ones introduced by Goodman in [13,14] for uniformly starlike and convex functions, except that in this case the point w is fixed. The functions f (z) in S w is said to be starlike functions of order β if and only if: for some ( ) We denote by S * w (β) the class of all starlike functions of order β.
Similarly, a functions f (z) in S w is said to be convex of order β if and only if: for some ( ) It follows from the definitions 3 and 4 that: We denote by C * w (β) the class of all convex functions of order β.
For the function f (z) in the class S w , we define: and for k = 1,2,3,... we can write: The differential operator I * studied extensively by [10,11] and in the case w = 0 was given by [9] .
Next, we will recall each of the following coefficient inequalities associated with the function classes S * w (k, β) and C * w (k, β) as well as some significant definitions which will contribute to this study.

Definitions and preliminaries:
Theorem A [11] if f∈S w , given by 2, satisfies the coefficient inequality: with ( ) Theorem B: If f S w ∈ , given by 2, satisfies the coefficient inequality: Proof: It is easy to check that if: Then we have * f C (k, ) w ∈ β . Hence the theorem.
In view of Theorem A and Theorem B, we now introduce the subclasses whose Taylor-Maclaurin coefficients a n satisfy the inequalities 3 and 4, respectively. In our proposed investigation of functions in the classes S * w (β) and C * w (β) we shall also make use of the following definitions and results.

Definition 1: (Hadamard Product or Convolution).
Given two functions f, g∈S w where f is given by 5 and g (z) is defined by: n ≥ ∈ The Hadamard product (or convolution) f*g is defined (as usual) by: The following constant factor in the subordination result (13): n c a z w n n z w n 1 Thus, by Definition 3, the subordination result 13 will hold true if: 1 a n 1 n 1 is a subordinating factor sequence (with, of course, a 1 = 1). In view of Theorem C, this is equivalent to the following inequality: where we have also made use of the assertion 7 of Theorem A. This evidently proves the inequality 17 and hence also the subordination result 13 asserted by Theorem 1.
The inequality 14 follows from 7 upon setting: z w 1 z w n z w z w n 1 Next we consider the function: which is a member of the class ( ) * w . S β Then, by using 13, we have: It is also easily verified for the function q (z) defined by 20 that: which completes the proof of Theorem 1.

Corollary:
Let the function f defined by 2 be in the class ( ).
ST w β Then the assertions 13 and 14 of Theorem 1 hold true. Furthermore, the following constant factor: 1 1 + β + β + α − αβ cannot be replaced by a larger one.
By taking α = 1 in the above corollary, we obtain.

Corollary:
Let the function f defined by 2 be in the class ( ) and ( ) ( ) The constant factor ( ) The following constant factor in the subordination result 25: