An estimation of the pre-Schwarzian norm for certain classes of analytic functions

TL;DR

This work addresses sharp bounds on the pre-Schwarzian norm $\|P_f\|$ for analytic functions in the Ma-Minda-type classes $\mathcal{S}^*_{hyp}$ and $\mathcal{S}^*_{L}$, along with their convex counterparts $\mathcal{C}_{hyp}$ and $\mathcal{C}_{L}$. The approach relies on representing $f$ via Schwarz functions tied to the Ma-Minda functions $\varphi_s(z)=1/(1-z)^s$ or $\varphi(z)=(1+s z)^2$ and applying logarithmic derivatives to bound $\|P_f\|$. The main results provide sharp bounds for each class in terms of a fixed $s$ and unique roots $t_s$ or $r_s$ of auxiliary equations, with explicit extremal functions $f_1$, $f_2$, $f_3$ achieving equality. These findings extend classical pre-Schwarzian estimates for univalent and starlike/convex subclasses and have implications for uniform local univalence in generalized Ma-Minda families.

Abstract

The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions $f$ in the class $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$ when $\varphi(z)=1/(1-z)^s$ with $0<s\leq 1$ and $\varphi(z)=(1+sz)^2$ with $0<s\leq 1/\sqrt{2}$, where $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $\varphi$, respectively.

Figures (5)

• Figure 1: Image of $\mathbb{D}$ under the mapping $1/(1-z)^s$ for $s=1/3$
• Figure 2: Image of $\mathbb{D}$ under the mapping $(1+sz)^2$ for $s=1/\sqrt{2}$
• Figure 3: Graph of $F_s(t)$ for different values of $s$ in $(0,1)$
• Figure 4: Graph of $G_s(t)$ for different values of $s$ in $(0,1/\sqrt{2}]$
• Figure 5: Graph of $h_s(r)$ for different values of $s$ in $(0,1)$

Theorems & Definitions (8)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof