SUBORDINATION BY CONVEX FUNCTIONS

For a fixed analytic function g ( z ) = z + ∑ n = 2 ∞ g n z n defined on the open unit disk and γ 1 , let T g ( γ ) denote the class of all analytic functions f ( z ) = z + ∑ n = 2 ∞ a n z n satisfying ∑ n = 2 ∞ | a n g n | ≤ 1 − γ . For functions in T g ( γ ) , a subordination result is derived involving the convolution with a normalized convex function. Our result includes as special cases several earlier works.


Introduction
Let Ꮽ be the class of all normalized analytic functions of the form a n z n z ∈ Δ := z ∈ C : |z| < 1 . (1.1) Let S * (α) and C(α) be the usual classes of normalized starlike and convex functions of order α, respectively, and let C := C(0).For f (z) given by (1.1) and g(z) by the convolution (or Hadamard product) of f and g, denoted by f * g, is defined by a n g n z n . (1. 3) The function f (z) is subordinate to the function g(z), written as f (z) ≺ g(z), if there is an analytic function w(z) defined on Δ with w(0 Let g(z) given by (1.2) be a fixed function, with g n ≥ g 2 > 0 (n ≥ 2), γ < 1, and let The class T g (γ) includes as its special cases various other classes that were considered in several earlier works.In particular, for γ = α and g n = n − α, we obtain the class TS * (α) := T g (γ) that was introduced by Silverman [6].Putting γ = α and g n = n(n − α), we get TC(α) := T g (γ).For these classes, Silverman [6] proved that TS * (α) ⊆ S * (α) and TC(α) ⊆ C(α).By using convolution, Ruscheweyh [5] defined the operator (1.5) Let R α (β) denote the class of functions f (z) in Ꮽ that satisfies the inequality Al-Amiri [1] called functions in this class as prestarlike functions of order α and type β.
Let H α (β) denote the class of functions f (z) given by (1.1) whose coefficients satisfy the condition where For functions in the class H α (β), Attiya [2] proved the following. (1.10) The constant factor in the subordination result (1.9) cannot be replaced by a larger number.
The constant factor in the subordination result (1.13) cannot be replaced by a larger number.
A similar result [8, Theorem 2, page 5] for ᏺ * (α) was also obtained.In this article, Theorems 1.1 and 1.2 are unified for the class T g (γ).Relevant connections of our results with several earlier investigations are also indicated.
We need the following result on subordinating factor sequence to obtain our main result.Recall that a sequence (b n ) ∞ 1 of complex numbers is said to be a subordinating factor sequence, if for every convex univalent function f (z) given by (1.1), then (1.16)

complex numbers is a subordinating factor sequence if and only if
(1.17)

Subordination with convex functions
We begin with the following subordination result.
The constant factor in the subordination result (2.1) cannot be replaced by a larger number.
, our result follows if we prove the result for the class T G (γ).Let f (z) ∈ T G (γ) and suppose that ( In this case, Observe that the subordination result (2.1) holds true if is a subordinating factor sequence (with of course, a 1 = 1).In view of Theorem 1.3, this is equivalent to the condition that ( (2.9) Clearly, F(z) ∈ T g (γ).For this function F(z), (2.1) becomes Therefore the constant g 2 2 g 2 + 1 − γ (2.12) cannot be replaced by any larger one.

.13)
The constant factor in the subordination result (2.13) cannot be replaced by a larger number.