Abstract
In this paper we introduce certain new subclasses of uniformly p-valent star like and convex functions. Sufficient coefficient conditions are obtained for functions in these classes. We provide geometrical properties of functions belonging to these classes. Hadamard product with convex functions and certain coefficient estimates are also obtained.
Highlights
Let Ap denote the class of functions f(z) of the form ∞∑ f(z) = z p + ak zk, p ∈ ={1, 2,....} which are analytic in the k= p+1 open unit disc U={z ∈ C : |z| < 1}.A function f ∈ Ap is said to be p−valent star like of order α (0 ≤ α
We introduce two new subclasses denoted by SDp (β,α) and KDp (β,α) of functions f(z) ∈ Ap as follows:
It is easy to see that for α=0, β=0 and p=1 in Theorem 5.1 and 5.2 we obtain well known coefficient estimates for functions in the classes of starlike and convex functions denoted by S* and K respectively [6, 7]
Summary
Let S*p (α) and Kp(α) denote, respectively, the classes of p-valent star like and convex functions of order α in U. Note that for p=1 the classes S*1 (α)=S⋆(α) and K1(α)=K(α) are the usual classes of univalent star like and univalent convex functions of order α (0 ≤ α < 1) respectively. We introduce two new subclasses denoted by SDp (β,α) and KDp (β,α) of functions f(z) ∈ Ap as follows:-. For p=1 the subclasses SD1(β,α)=SD(β,α) and KD1(β,α)=KD (β,α) were introduced and studied by Shams, Kulkarni and Jahangiri in [1] They obtained sufficient coefficient conditions for functions in the classes SD(β,α) and KD(β,α) along with geometric properties of functions in these classes. In this paper we shall study the geometric properties, coefficient bounds and convolution properties for functions in the classes SDp(β,α) and KDp(β,α). We will show that these classes are closed under certain integral operators
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