Abstract
In this paper, we define and study some new subclasses of starlike and close-to-convex functions with respect to symmetrical points. These functions map the open unit disc onto certain conic regions in the right half plane. Some basic properties, a necessary condition, and coefficient and arc length problems are investigated. The mapping properties of the functions in these classes are studied under a certain linear operator.
Highlights
Let A be the class of functions of the form ∞f (z) = z + anzn, ( . ) n=which are analytic in the open unit disc E = {z : |z| < }
Let S, K, S∗, and C be the subclasses of A which consist of univalent, close-to-convex, starlike, and convex functions, respectively
A function f in A is said to be uniformly convex in E if f is a univalent convex function along with the property that, for every circular arc γ contained in E, with center ξ in E, the image curve f (γ ) is a convex arc
Summary
Sakaguchi [ ] introduced and studied the class Ss∗ of starlike functions with respect to symmetrical points. Das and Singh [ ] defined the classes Cs of convex functions with respect to symmetrical points and showed that a necessary and sufficient condition for f ∈ Cs is that (zf (z)) ∈ P, z ∈ E. Ss∗, we note that k – UKs(β) ⊂ Ks ⊂ K , where kS consists of close-to-convex functions with respect to symmetrical starlike functions.
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