On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

Mohsan Raza,Sahidul Islam,Sarfraz Nawaz Malik,Hira Naz,Fairouz Tchier

doi:10.1155/2021/5353372

Abstract

This article comprises the study of differential subordination with analogue of q -derivative. It includes the sufficient condition on γ for 1 + γ ∂ z q h z / h n z to be subordinated by 1 + A z / 1 + B z , − 1 ≤ B < A ≤ 1 , and implies that h z ≺ 1 + z , where h z is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

Highlights

N 2 as Taylor series. e class S comprises the normalized univalent functions, defined in U. e major subcategories of class S are C of convex functions and S∗ of starlike functions
Subordination plays an important role in univalent function theory, and this concept was first introduced by Lindelof, but Littlewood [3, 4] contributed remarkably to this field
E advancement in the field of differential subordination starts with the usage of univalent functions

Summary

Introduction

N 2 as Taylor series. e class S comprises the normalized univalent functions, defined in U. e major subcategories of class S are C of convex functions and S∗ of starlike functions. Let a set A be considered as the class of analytic functions defined in open unit disk U {ς: ς ∈ C and |ς| < 1} under normalization conditions f(0) 0 and f′(0) 1, having f(ς) ς + 􏽘 anςn, ς ∈ U, (1) Subordination plays an important role in univalent function theory, and this concept was first introduced by Lindelof, but Littlewood [3, 4] contributed remarkably to this field. Many researchers contributed in the work related to differential subordinations.

Results

Conclusion