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The modified Fekete-Szegö functional for a subclass of close-to-starlike mappings in complex Banach spaces

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Abstract

In [23], Koepf proved that for a function \(f(\xi )=\xi +\sum \limits _{m=2}^\infty a_m\xi ^m\) in the class of normalized close-to-convex functions in the unit disk,

$$\begin{aligned} |a_3-\lambda a_2^2|\le \left\{ \begin{array}{ll} 3-4\lambda ,\quad &{} \lambda \in [0, \frac{1}{3}],\\ \frac{1}{3}+\frac{4}{9\lambda },\quad &{} \lambda \in [\frac{1}{3}, \frac{2}{3}],\\ 1,\quad &{} \lambda \in [\frac{2}{3}, 1]. \end{array}\right. \end{aligned}$$

In this paper, considering the zero of order (i.e., the mapping \(f(x)-x\) has zero of order \(k+1\) at the point \(x=0\)), we generalize the above classical result and establish the modified Fekete-Szegö functional for s subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.

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Acknowledgements

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

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Authors and Affiliations

  1. School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    Qinghua Xu & Huihui Li

  2. Department of Mathematics, Huzhou University, Huzhou, 313000, People’s Republic of China

    Taishun Liu

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  1. Qinghua Xu
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Correspondence to Qinghua Xu.

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

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Xu, Q., Li, H. & Liu, T. The modified Fekete-Szegö functional for a subclass of close-to-starlike mappings in complex Banach spaces. Anal.Math.Phys. 14, 45 (2024). https://doi.org/10.1007/s13324-024-00910-5

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