Starlikeness of certain non-univalent functions

Adam Lecko,V Ravichandran,Asha Sebastian

doi:10.1007/s13324-021-00600-6

Abstract

We consider three classes of functions defined using the class {mathcal {P}} of all analytic functions p(z)=1+cz+cdots on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with f/gin {mathcal {P}} and g/(zp)in {mathcal {P}} for some normalized analytic function g and pin {mathcal {P}}. The second class is defined by replacing the condition f/gin {mathcal {P}} by |(f/g)-1|<1 while the other class consists of normalized analytic functions f with f/(zp)in {mathcal {P}} for some pin {mathcal {P}}. We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order alpha , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

Highlights

Let D denote the open unit disc in C
Several well known subclasses of starlike and convex functions were characterized by subordination of z f (z)/ f (z) or 1 + (z f (z)/ f (z) to some function in P
Univalent in unit disc D with Re φ(z) > 0, starlike with respect to φ(0) = 1, symmetric about the real axis and φ (0) > 0, Ma and Minda [22] gave a unified treatment for several subclasses of starlike and convex functions by studying the classes S∗(φ) = { f ∈ A : z f (z)/ f (z) ≺ φ(z)} and K(φ) = { f ∈ A : 1 + z f (z)/ f (z) ≺ φ(z)}

Summary

Introduction

Let D denote the open unit disc in C. Univalent in unit disc D with Re φ(z) > 0, starlike with respect to φ(0) = 1, symmetric about the real axis and φ (0) > 0, Ma and Minda [22] gave a unified treatment for several subclasses of starlike and convex functions by studying the classes S∗(φ) = { f ∈ A : z f (z)/ f (z) ≺ φ(z)} and K(φ) = { f ∈ A : 1 + z f (z)/ f (z) ≺ φ(z)}.

The class G2

The class G3

Radius of starlikeness