Some refinements of the Fekete–Szegö inequalities for a class of biholomorphic mappings

Abstract

Let $\mathbb{B}$ be the unit disk $\mathbb{U}$ in $\mathbb{C}$ or the unit ball of a general complex Banach space. In this paper, we obtain a refinement of the Fekete–Szegö inequality for the class of $S_{k+1}^0(\mathbb{B})$ of mappings f for which there exist g-Loewner chains $f(z,t)$, where $g(\xi )=\frac{1+\xi }{1-\xi },\xi \in \mathbb{U}$, such that $f(z)=f(z,0)$ and z = 0 is a zero of order k + 1 of $e^{-t}f(z,t)-z$ for each $t\ge 0$. The results presented generalize the classical Fekete–Szegö inequality and the results in Hamada et al. (Fekete–Szegö problem for univalent mappings in one and higher dimensions. J Math Anal Appl. 2022;516:126526).

Keywords:

• Fekete–Szegö inequality
• Loewner chain
• generating vector field

AMS Subject Classifications:

• Primary 32H02
• Secondary 30C45

Acknowledgments

The authors would like to thank the referees for useful suggestions which improved the paper.

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Disclosure statement

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

Additional information

Funding

This work was supported by NNSF of China (Grant No. 11971165) and Natural Science Foundation of Zhejiang Province (Grant No. LY21A010003).