Abstract
A new subclass Kp(α1,α2,β) of p-valent close-to-convex mappings defined by two-sided inequality is introduced. Some sufficient conditions for functions to be in Kp(α1,α2,β) are given.
Highlights
Let A(p) be the class of functions of the form ∞f (z) = zp + ∑ap+nzp+n (p ∈ N) (1)n=1 which are p-valent analytic in the open unit disk U = {z ∈ C : |z| < 1}
There and in the following, let N, C, and R be the sets of positive integers, complex numbers, and real numbers, respectively
A function f analytic in U is said to be close-toconvex if there is a convex function g such that for all z ∈ U
Summary
A number of results for close-to-convex functions in U have been obtained by several authors (see, e.g., [2–14]). A function f ∈ A(p) is said to be in the class Kp(α1, α2, β) if it satisfies the following two-sided inequality: A function f ∈ Kp(α, α, β) is called p-valent close-to-convex of order α and type β in U. 1−z is analytic and univalent convex in U and g (U)
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