Some refinements of the Fekete and Szegö inequalities in one and higher dimensions

Abstract

In this paper, we introduce a class of holomorphic functions ${\mathcal{M}}_g\left(\mathbb{U}\right)$ on unit disk $\mathbb{U}$. Let $w\left(\xi \right)$ be a normalized holomorphic function on $\mathbb{U}$ such that $w\left(\xi \right)/w^{\prime }\left(\xi \right)\in {\mathcal{M}}_g\left(\mathbb{U}\right)$, and $w\left(\xi \right)-\xi$ has a zero of order k + 1 at $\xi =0$. By using the formula of Faà di Bruno for the kth derivative of a composite function, we establish the Fekete and Szegö inequality for $w\left(\xi \right)$, and then we extend this result to higher dimensions. The results presented here generalize some known results. Finally, a certain problem is also considered.

Keywords:

• Fekete and Szegö inequality
• a class of holomorphic mappings
• sharp coefficient bound

AMS Subject Classifications:

• Primary 32H02
• Secondary 30C45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China (NNSF) of China (Grant Number 11971165).