STARLIKENESS AND CONVEXITY OF A CLASS OF ANALYTIC FUNCTIONS

Let be the class of analytic functions in the unit disk that are normalized with f (0)= f ′(0)− 1 = 0 and let −1 ≤ B < A ≤ 1. In this paper we study the class Gλ,α = { f ∈ : |(1− α + αz f (z)/ f ′(z))/z f ′(z)/ f (z)− (1− α)| < λ, z ∈ }, 0 ≤ α ≤ 1, and give sharp sufficient conditions that embed it into the classes S∗[A,B] = { f ∈ : z f ′(z)/ f (z) ≺ (1 +Az)/(1 + Bz)} and K(δ) = { f ∈ : 1 + z f (z)/ f ′(z) ≺ (1− δ)(1 + z)/(1− z) + δ}, where “≺ ” denotes the usual subordination. Also, sharp upper bound of |a2| and of the Fekete-Szegö functional |a3−μa2| is given for the class Gλ,α.


Introduction and preliminaries
A region Ω from the complex plane C is called convex if for every two points ω 1 ,ω 2 ∈ Ω, the closed line segment [ω 1 ,ω 2 ] = {(1 − t)ω 1 + tω 2 : 0 ≤ t ≤ 1} lies in Ω. Fixing ω 1 = 0 brings the definition of starlike region.If Ꮽ denotes the class of functions f (z) that are analytic in the unit disk ᐁ = {z : |z| < 1} and normalized by f (0) = f (0) − 1 = 0, then a function f ∈ Ꮽ is called convex or starlike if it maps ᐁ into a convex or starlike region, respectively.Corresponding classes are denoted by K and S * .It is well known that K ⊂ S * , and it is well known that both are subclasses of the class of univalent functions and have the following analytical representations: (1.1) More about these classes may be found in [2].Further, let f ,g ∈ Ꮽ.Then we say that f (z) is subordinate to g(z), and we write f (z) ≺ g(z), ifthere exists a function ω(z), analytic in the unit disk ᐁ, such that ω(0 In terms of subordination, we have These classes are widely studied during the past decades, mainly in two different directions: for developing criteria for starlikeness or convexity and for obtaining properties of the Maclaurin coefficients of a starlike or convex function.In this paper sufficient conditions (some of them sharp) that embed the class 0 < α ≤ 1, λ > 0, into the classes S * [A,B] and K(δ), 0 ≤ δ < 1, will be given, together with sharp upper bound of the Fekete-Szegö functional |a 3 − μa 2 2 |, μ ∈ R. Sufficient motivation for studying the class G λ,α is the fact that it makes close connection between classes, studied in [1,[6][7][8][9][11][12][13] and other references.

Conditions for starlikeness and convexity
For obtaining the result for convexity and starlikeness of the class G λ,α , we will use the method of differential subordinations.Valuable reference on this topic is [5].The general theory of differential subordinations, as well as the theory of first-order differential N. Tuneski and H. Irmak 3 subordinations, was introduced by Miller and Mocanu in [3,4].Namely, if φ : The univalent function q(z) is said to be a dominant of the differential subordination (2.1) if p(z) ≺ q(z) for all p(z) satisfying (2.1).If q(z) is a dominant of (2.1) and q(z) ≺ q(z) for all dominants of (2.1), then we say that q(z) is the best dominant of the differential subordination (2.1).
From the theory of first-order differential subordinations, we will make use of the following lemma.
In the beginning, using Lemma 2.1 we will prove the following result.
and φ(ω) are analytic in the domain D = C which contains q(ᐁ) and φ(ω) when ω ∈ q(ᐁ).Further, The result is sharp as the functions ze Az and z(1 + Bz) A/B show in the cases B = 0 and B = 0, respectively.Remark 2.3.According to the definition of subordination, the sharpness of the result of Theorem 2.2 means that h(ᐁ) is the greatest region in the complex plane with the property that if The following corollary gives sharp sufficient conditions that embed G λ,α into S * [A,B].
is the greatest number such that G λ,α ⊆ S * [A,B].
Proof.In order to prove this corollary, due to Theorem 2.2 it is enough to show that where h(z) is defined as in the statement of the theorem and (2.9) Further, let ) The last equality holds because 1 If α > 1/2, we have the following analysis.Equation ψ t (t) = 0 has unique solution (2.12) N. Tuneski and H. Irmak 5 It can be verified that |t * | > 1 is equivalent to At the end, the function has the opposite sign of the sign of coefficient A. Therefore, Sharpness of the result follows from the sharpness of Theorem 2.2 (see Remark 2.3) and the fact that the obtained λ is the greatest, which embeds the disk The following example exhibits some concrete conclusions that can be obtained from the results of the previous section by specifying the values α, A, B.
The value of λ in each of the above cases is the greatest that makes the corresponding inclusion true.Next theorem studies connection between G λ,α and the class of convex functions of some order.

Sharp estimate of the Fekete-Szegö functional
In this section we give sharp estimates of |a 2 | and of the Fekete-Szegö functional |a 3 − μa 2  2 | for a function f ∈ G λ,α .We will use following lemmas.
n be an analytic function in the unit disk ᐁ and |ω