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Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes

Raju Biswas

TL;DR

This work derives sharp estimates for the pre-Schwarzian norm ∥P_f∥ of functions in the Ma–Minda-type starlike and convex classes, focusing on three specific φ choices: φ(z) = 3/(3+ (α−3)z − α z^2), φ(z) = (1+z)(1−s z), and φ(z) = 1+ z/((1−z)(1+α z)). By expressing zf'(z)/f(z) or 1+zf''(z)/f'(z) via subordination to φ and applying Schwarz–Pick bounds, the authors reduce the problem to optimizing auxiliary functions over radial parameters, yielding explicit sharp bounds such as ∥P_f∥ ≤ 2(α+6)/(3+α) for S^*_{con} and ∥P_f∥ ≤ 2/(1+α) for C_{cs}, among others. The results are proved sharp with concrete extremal functions, and the analysis covers both the starlike and convex Ma–Minda families, including cases where the extremal roots t_s and r_s are characterized as unique solutions of auxiliary equations. Overall, the paper advances the understanding of how Ma–Minda data control the geometric behavior of f via the pre-Schwarzian norm, with potential implications for univalence criteria and Teichmüller-theoretic considerations.

Abstract

In this paper, we establish the sharp estimates of the pre-Schwarzian norm of functions $f$ in the Ma-Minda type starlike and convex classes $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$, respectively, whenever $\varphi(z)=3/\left(3+(α-3)z-αz^2\right)$ with $-3<α\leq 1$, $\varphi(z)=(1+z)(1-s z)$ with $-1/3\leq s\leq 1/3$ and $\varphi(z)=1+z/\left((1-z) (1+αz)\right)$ with $0\leq α\leq 1/2$.

Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes

TL;DR

This work derives sharp estimates for the pre-Schwarzian norm ∥P_f∥ of functions in the Ma–Minda-type starlike and convex classes, focusing on three specific φ choices: φ(z) = 3/(3+ (α−3)z − α z^2), φ(z) = (1+z)(1−s z), and φ(z) = 1+ z/((1−z)(1+α z)). By expressing zf'(z)/f(z) or 1+zf''(z)/f'(z) via subordination to φ and applying Schwarz–Pick bounds, the authors reduce the problem to optimizing auxiliary functions over radial parameters, yielding explicit sharp bounds such as ∥P_f∥ ≤ 2(α+6)/(3+α) for S^*_{con} and ∥P_f∥ ≤ 2/(1+α) for C_{cs}, among others. The results are proved sharp with concrete extremal functions, and the analysis covers both the starlike and convex Ma–Minda families, including cases where the extremal roots t_s and r_s are characterized as unique solutions of auxiliary equations. Overall, the paper advances the understanding of how Ma–Minda data control the geometric behavior of f via the pre-Schwarzian norm, with potential implications for univalence criteria and Teichmüller-theoretic considerations.

Abstract

In this paper, we establish the sharp estimates of the pre-Schwarzian norm of functions in the Ma-Minda type starlike and convex classes and , respectively, whenever with , with and with .
Paper Structure (3 sections, 6 theorems, 113 equations, 5 figures, 2 tables)

This paper contains 3 sections, 6 theorems, 113 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Pre-Schwarzian Norm
  3. Main results

Key Result

Theorem 3.1

Let $f\in\mathcal{S}^*_{con}$. Then the pre-Schwarzian norm satisfies the following sharp inequality

Figures (5)

  • Figure 1: Image of $\mathbb{D}$ under the mapping $3/\left(3+(\alpha-3)z-\alpha z^2\right)$, $\alpha=-1$
  • Figure 2: Image of $\mathbb{D}$ under the mapping $(1+z)(1-s z)$, $s=1/3$
  • Figure 3: Image of $\mathbb{D}$ under the mapping $1+z/\left((1-z) (1+\alpha z)\right)$, $\alpha=1/3$
  • Figure 4: Graph of $H_1(\alpha)$ and $H_2(\alpha)$ for $\alpha$ in $[-3/2,0)$
  • Figure 5: Graph of $H_s(t)$ for different values of $s$ in $[-1/3, 1/3]$

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 2 more