Some Further Properties for Analytic Functions with Varying Argument Defined by Hadamard Products

The purpose of this paper is to obtain some further properties including coefficients estimates, majorization problems, distortion bounds, extreme points and radius of close-to-convexity, starlikeness and convexity for functions belonging to the class TUγ(φ, ψ;α,A,B), which are defined by Hadamard products with varying argument. Mathematics Subject Classification: 30C45; 30C50; 26D15


Introduction
Let A denote the class of functions of the form which are analytic in the open unit disc U = {z ∈ C : |z| < 1}.Let S be the subclass of A, consisting of analytic and univalent functions.We denote by S * (β) and K(β) (0 ≤ β < 1) the class of starlike of order β in U and the class of convex functions of order β in U, respectively.It is well known that S * (β) ⊂ S * (0) = S * and K(β) ⊂ K(0) = K.
A function f (z) ∈ A is said to be in U S(α, β), the class of α-uniformly starlike functions of order β (0 ≤ β < 1), if f satisfies the condition (see [1,2]) Replacing f (z) in (1.2) by zf (z), we obtain Required for the function f to be in the subclass U K(α, β) of α-uniformly convex functions of order β.
Also, by T γ (γ ∈ R) we denote the class of functions f (z) ∈ A of the form (1.1) for which all of non-vanishing coefficients satisfy the condition arg(a n ) = π + (1 − n)γ (n = 2, 3, • • •). (1.4) For γ = 0 we obtain the class T 0 of functions with negative coefficients.Moreover, we define The class T was introduced by Silverman [3] (see also [4], [5] and [31]).It is called the class of functions with varying argument of coefficients.
For two functions f and g, analytic in U, we say that the function f is subordinate to g in U, and write f (z) ≺ g(z) (z ∈ U), if there exists a Schwarz function ω, which (by definition) is analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈ U), such that f (z) = g(ω(z)) (z ∈ U).Furthermore, if the function g is univalent in U, then we have the following equivalence [6, p.4]: Let f and g be analytic in the open unit disk U. We say that f is majorized by g in U (see [7]) and write if there exists a function ϕ(z), analytic in U such that It may be noted here that (1.5) is closely related to the concept of quasi-subordination between analytic functions.
For arbitrary fixed real numbers A and B (−1 ≤ B < A ≤ 1), let P (A, B) denote the class of functions of the form p(z) = 1 + ∞ j=1 p j z j , which are analytic in U and satisfies the condition The class P (A, B) was introduced and studied by Janowski [8].We note that a function f (z) ∈ P (A, B) if and only if (1.7) Let For α ≥ 0, −1 ≤ B < A ≤ 1 and for all z ∈ U, Li et al. [9] defined the subclass U (φ, ψ, α, A, B) of A which satisfies the following condition: where are analytic in U such that (f * ψ)(z) = 0 and µ j > η j ≥ 0 for j ≥ 2.
For α = 0, A = 1 − 2β (0 ≤ β < 1) and B = −1, we denote the class U (φ, ψ, 0, 1 − 2β, −1) by U (φ, ψ; β).Equivalently, U (φ, ψ; β) can be expressed in the form Using the results in [10,11,12] and (1.8), we get the following geometric interpretation.Geometric interpretation.f (z) ∈ U (φ, ψ; α, A, B) if and only if p(z) = (f * φ)(z) (f * ψ)(z) takes all values in the conic domain R α (A, B) which is included in the right half plane such that Denote by P(p(α, A, B)), the family of functions p, such that p ∈ P, where P denotes the well-known class of Caratheodory functions and p ≺ p(α, A, B)(z) in U.The function p(α, A, B)(z) maps the unit disk conformally onto the domain R α (A, B) such that 1 ∈ R α (A, B) and ∂R α (A, B) is a curve defined by the equality From elementary computations, we see that (1.10) represents conic sections symmetric about the real axis.Thus R α (A, B) is an elliptic domain for α > 1, a parabolic domain for α = 1, a hyperbolic domain for 0 < α < 1 and the right half plane for α = 0.The functions which play the role of extremal functions for these conic regions are given as where , every positive number α can be expressed as α = cosh πh (t) 4h(t) , where h(t) is the Legendre's complete elliptic integral of the first kind and h (t) is complementary integral of h(t) (for details, see [10,11,12]).Also from (1.11), we get the extremum function p(α, A, −1)(z) (B = −1).
In this paper, we aim to obtain some further properties, such as coefficients estimates, majorization problems, distortion bounds, extreme points and radius of close-to-convexity, starlikeness and convexity for functions belonging to the class T U γ (φ, ψ; α, A, B).

Preliminary results
We need the following results in our next investigation.

Main Results
Proof.Suppose that f ∈ U (φ, ψ; α, A, B).Then, by Lemma 2.1, we obtain Let us define the function p(z) by Hence p(z) is analytic in U with p(0) = 1 and p(z) > 0 (z ∈ U ).Let
So (3.2) is true for j = m + 1.Consequently, using the mathematical induction, we get that (3.2) holds true for all j ≥ 3.
where r 0 = r 0 (α, A, B) is the smallest positive root of the equation Proof.Suppose that f ∈ T U γ (φ, ψ; α, A, B).Then, by Lemma 2.3, we obtain or, equivalently, which holds true for all z ∈ U. We find from (3.6) that where ω(z) = c 1 z + c 2 z 2 + • • • ∈ W, W denotes the well known class of the bounded analytic functions in U and satisfies the conditions: From (3.7), we get Next, since (f * φ)(z) is majorized by (f * ψ)(z) in U, from (1.6), we have Differentiating it with respect to z and multiplying by z, we get Thus, by Lemma 2.4, (3.8) and (3.9), we get where takes its maximum value at ρ = 1, with r 0 = r 0 (α, A, B), where r 0 = r 0 (α, A, B) is the smallest positive root of (3.5).Furthermore, if 0 ≤ δ ≤ r 0 (α, A, B), then the function ψ(ρ) defined by is an increasing function on the interval 0 ≤ ρ ≤ 1, so that Hence, upon setting ρ = 1 in (3.13), we conclude that (3.4) of Theorem 3.2 holds true for |z| ≤ r 0 = r 0 (α, A, B), which completes the proof of Theorem 3.2.
where r 0 = r 0 (A, B) is the smallest positive root of the equation Proof.Suppose that f ∈ T U γ (φ, ψ; 1, A, B).Then, by Lemma 2.5, we obtain or, equivalently, which holds true for all z ∈ U. We find from (3.17 where A j and B j are given by (2.8) and (2.10), respectively.