Subordination Results for Certain Subclasses of Uniformly Starlike and Convex Functions Defined by Convolution
In this paper we derive several subordination results for certain subclasses of uniformly starlike and convex functions defined by convolution.
Classes of uniformly starlike and convex functions
We note that for α> 1, if f ∈ SD(α, β) ,t henzf � (z)/f (z) lies in the region G ≡ G(α, β) ≡ {w :R e w> α|w −1|+β}, that is, part of the complex plane which contains w = 1a nd is bounded by the ellipse (u−(α 2 −β)/(α 2 −1)) 2 +(α 2 /(α 2 −1))v 2 = α 2 (1 − β) 2 /(α 2 − 1) 2 with vertices at the points ((α + β)/(α + 1), 0), ((α − β)/(α − 1), 0), ((α 2 − β)/(α 2 − 1), (β − 1)/ √ α 2 − 1), and ((α 2 − β)/(α 2 − 1), (1 − β)/ √ α 2 − 1) .S ince β β } and so SD(α, β) ⊂ S ∗ (β) .F orα = 1 if f ∈ SD(α, β) ,t henzf � (z)/f (z) belongs to the region which contains w = 2 and is bounded by parabola u = (v 2 + 1 − β 2 )/2(1 − β). Using the relation between convex and starlike functions, we define KD(α, β) as the class of functions f ∈ A if and only if zf � ∈ SD(α, β) .F orα = 1 and β = 0, we obtain the class KD(1, 0) of uniformly convex functions, first defined by Goodman [1]. Ronning [3] investigated the class KD(1 ,β )of uniformly convex functions of order β. For the class KD(α, 0) of α-uniformly convex function, see [2]. In this note, we study the coefficient bounds and Hadamard product or convolution properties of the classes SD(α, β) and KD(α, β). Using these results, we further show that the classes SD(α, β) and KD(α, β) are closed under certain integral operators. 2. Main results. First we give a sufficient coefficient bound for functions in SD(α, β). Theorem 2.1. If ∞=2[n(1 + α)− (α + β)]|an| < 1 − β, then f ∈ SD(α, β).
APPLICATIONS OF SRIVASTAVA-ATTIYA OPERATOR TO THE CLASSES OF STRONGLY STARLIKE AND STRONGLY CONVEX FUNCTIONS
Srivastava-Attiya operator is used to define some new subclasses of strongly starlike and strongly convex functions of order β and type α in the open unit disk U. For each of these new function classes, several inclusion relationships are established. Some interesting corollaries and applications of the results presented here are also discussed.