On the Inequality Theorem for a Wider Class of Analytic Functions with Hadamard Product

In this paper, we discuss a well known class studied by Ramesha in 1995 and later by Janteng in 2006, and we then extend the class to a wider class of functions f denoted by n α which are normalized and univalent, in the unit diskD z ∈ C: |z|< 1 { } satisfying the condition Re((αz2f′′(z)/g(z)) + (zf′(z)/g(z)))> β, 0≤ α< 1, 0≤ β< 1,where g is analytic function inD, such that g(z)≠ 0, with a new condition that is introduced. e main purpose of this paper is to give an estimate for the same |a3 − μa2| when f belongs to the class n α .


Introduction and Definition
Let S denote the class of normalized analytic univalent functions f of the form f(z) z + ∞ n 2 a n z n , a n is complex number, where z ∈ D z: |z| < 1 { } . Also, let g(z) z + ∞ n 2 b n z n , ϕ(z) z + ∞ n 2 k n z n , and ψ(z) z + ∞ n 2 d n z n be analytic functions in D where b n , k n , d n > 0 and k n > d n . We de ne the Hadamard product as follows: g(z) * ϕ(z) z + (2) Earlier in 1933, Fekete and Szego [1] states that for f ∈ S and given by (1), for 0 ≤ μ ≤ 1 and the inequality is sharp. e Fekete-Szego problems for the subclass of S consisting of the families, convex functions C, starlike functions S * , and close-to-convex functions CC have been completely solved in the literature. Among others are Keogh and Merkes [2], Koepf [3], Darus and omas [4,5], Frasin and Darus [6], Ebadian et al. [7], and Mohammed et al. [8]. In particular, for f ∈ CC and be given by (1), Keogh and Merkes [2] showed that and for each μ, there is a function in CC for which equality holds.
Moreover, an estimate is given for the same functional for the new class n β,c α defined as follows.
Definition 1. For 0 ≤ α < 1, 0 ≤ β < 1, and 0 ≤ c < 1, let the function f be given by (1). en, the function f ∈ n β,c α if and only if there exists g analytic function, g(z) ≠ 0, such that for z ∈ D, Re Re is class is extended from Ramesha et al. [10] and Janteng [11], for suitable choices of ϕ, ψ, we easily obtain the various subclasses of S. (6), we get the class of convex functions g of order c(0 ≤ c < 1) denoted by C(c).
To establish our main theorems in this paper, we first state some preliminary lemmas, required in proving of our theorem.

Preliminary Results
Lemma 1 (see [9]). Let h be analytic in D with Re h(z) > 0 and be given by Lemma 2 (see [2]). Let g ∈ S * , the starlike function with e first result for the class n β,c α is as follows.

Theorem 1.
Let the function f be given by (1) and belong to the class n β,c α . en, for 0 ≤ α < 1, e inequalities are sharp for all cases. However, the proofs for the case μ � μ 2 are still unsolved.

International Journal of Mathematics and Mathematical Sciences
From (20), we have Now, consider the first case for all that is, having b 3 − 3(2α + 1)μb 2 2 /4(α + 1) 2 > 0 and 2(α + 1) 2 − 3(2α + 1)μ > 0. Inequalities and follow immediately from Lemma 1. Now we get Now, after doing some operations, the function ϕ achieves its maximum value at Next, we find Finally, we get our results: Now since we know | x°| ≤ 2, we get the interval Hence, result (28) concludes for the case