Abstract
In this paper, our aim is to define certain new classes of multivalently spiral-like, starlike, convex and the varied Mocanu-type functions, which are associated with conic domains. We investigate such interesting properties of each of these function classes, such as (for example) sufficiency criteria, inclusion results and integral-preserving properties.
Highlights
Introduction and MotivationLet A( p) denote the class of functions of the form: f (z) = z p + ∞∑ an+ p zn+ p ( p ∈ N = {1, 2, 3, · · · }), (1)n =1 which are analytic and p-valent in the open unit disk:E = {z : z ∈ C and |z| < 1}.In particular, we write: A(1) = A.by S ⊂ A, we shall denote the class of all functions that are univalent in E.The familiar class of p-valently starlike functions in E will be denoted by S ∗ ( p), which consists of functions f ∈ A( p) that satisfy the following conditions:
We denote by K the class of close-to-convex functions, which consists of functions f ∈ A that satisfy the following inequality:
Using the idea of spiral-like and close-to-convex functions, we have introduced Mocanu-type functions associated with conic domains
Summary
Let A( p) denote the class of functions of the form:. n =1 which are analytic and p-valent in the open unit disk:. In the year 1967, Libera [3] extended this definition to the class of functions, which are spiral-like of order ρ denoted by Sρ∗ ( β) as follows. In this paper, we define certain new subclasses of spiral-like close-to-convex functions by using the idea of Noor et al [6] and Umarani [5]. By using these conic domains Ωk (k = 0), they introduced and studied the corresponding class k-ST of k-starlike functions (see Definition 3 below). Motivated and inspired by the recent and current research in the above-mentioned work, we here introduce and investigate certain new subclasses of analytic and p-valent functions by using the concept of conic domains and spiral-like functions as follows.
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