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Second and Third order differential subordination for exponential function

S. Sivaprasad Kumar, Neha Verma

Abstract

This article presents several findings regarding second and third-order differential subordination of the form: $$ p(z)+γ_1 zp'(z)+γ_2 z^2p''(z)\prec h(z)\implies p(z)\prec e^z $$ and $$ p(z)+γ_1 zp'(z)+γ_2 z^2p''(z)+γ_3 z^3p'''(z)\prec h(z)\implies p(z)\prec e^z. $$ Here, $γ_1$, $γ_2$, and $γ_3$ represent positive real numbers, and various selections of $h(z)$ are explored within the context of the class $\mathcal{S}^{*}_{e} := \{f \in \mathcal{A} : zf'(z)/f(z) \prec e^z\}$, which denotes the class of starlike functions associated with the exponential function.

Second and Third order differential subordination for exponential function

Abstract

This article presents several findings regarding second and third-order differential subordination of the form: and Here, , , and represent positive real numbers, and various selections of are explored within the context of the class , which denotes the class of starlike functions associated with the exponential function.
Paper Structure (4 sections, 12 theorems, 63 equations, 1 figure)

This paper contains 4 sections, 12 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Second order differential subordination
  4. Third order differential subordination

Key Result

Lemma 1.2

antoninoandmiller Let $z_0\in \mathbb{D}$ and $r_0=|z_0|$. Let $f(z)=a_nz^n+a_{n+1}z^{n+1}+\cdots$ 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)|=\ma 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 (solid green) represents $z+\sqrt{1+z^2}$.

Theorems & Definitions (22)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Lemma 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 12 more