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Higher order differential subordinations for certain starlike functions

Neha Verma, S. Sivaprasad Kumar

Abstract

In this paper, we employ a novel second and third-order differential subordination technique to establish the sufficient conditions for functions to belong to the classes $\mathcal{S}^*_s$ and $\mathcal{S}^*_ρ$, where $\mathcal{S}^*_s$ is the set of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec 1+\sin z$ and $\mathcal{S}^*_ρ$ is the set of all normalized analytic functions $f$ satisfying $ zf'(z)/f(z)\prec 1+\sinh^{-1} z$.

Higher order differential subordinations for certain starlike functions

Abstract

In this paper, we employ a novel second and third-order differential subordination technique to establish the sufficient conditions for functions to belong to the classes and , where is the set of all normalized analytic functions satisfying and is the set of all normalized analytic functions satisfying .
Paper Structure (3 sections, 34 theorems, 127 equations, 1 figure, 1 table)

This paper contains 3 sections, 34 theorems, 127 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The Class $\mathcal{S}^{*}_s$
  3. The Class $\mathcal{S}^{*}_{\rho}$

Key Result

Lemma 1.2

antoninoandmiller Let $z_0\in \mathbb{D}$ and $r_0=|z_0|$. Let $f(z)=\sum_{k=n}^{\infty}a_kz^k$ be continuous on $\overline{\mathbb{D}}_{r_0}$ and analytic on $\mathbb{D}\cup\{z_0\}$ with $f(z)\neq 0$ and $n\geq 2$. If $|f(z_0)|=\max \{|f(z)|:z\in \overline{\mathbb{D}}_{r_0}\}$ and $|f'(z_0)|=\max\{ where $l\geq k\geq m\geq n\geq2$.

Figures (1)

  • Figure 1: Graph of two circles, namely $C_1$ (blue boundary) and $C_2$ (orange boundary). While the shaded region represents $z+\sqrt{1+z^2}$.

Theorems & Definitions (60)

  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Definition 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Lemma 1.7
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • ...and 50 more