Inclusion relations for certain families of integral operators associated with conic regions

Shahid Mahmood,Hari Mohan Srivastava,Imran Khan,Sarfraz Nawaz Malik

doi:10.1186/s13660-019-2015-9

Abstract

In this work, we introduce certain subclasses of analytic functions involving the integral operators that generalize the class of uniformly starlike, convex, and close-to-convex functions with respect to symmetric points. We then establish various inclusion relations for these newly defined classes.

Highlights

Let A be the class of functions ∞f (z) = z + anzn n=2 (1.1)analytic in the open unit disc A = {z ∈ C : |z| < 1}, and let S be the class of functions in A that are univalent in A
Analytic in the open unit disc A = {z ∈ C : |z| < 1}, and let S be the class of functions in A that are univalent in A
When g is univalent, such a subordination is equivalent to f (0) = g(0) and f (A) ⊆ g(A); see [1]

Summary

Introduction

1 Introduction Let A be the class of functions Analytic in the open unit disc A = {z ∈ C : |z| < 1}, and let S be the class of functions in A that are univalent in A. Let S∗, C, K, and C∗ be the subclasses of A consisting of all functions that are starlike, convex, close-to-convex, and quasiconvex, respectively; for details, see [1]. Sakaguchi [2] introduced and studied the class Ss∗ of starlike functions with respect to symmetrical points z and –z belonging to the open unit disc A.

Results

Conclusion