Abstract
Let \(\mathcal {S}\) denote the class of analytic and univalent (i.e., one-to-one) functions \( f(z)= z+\sum _{n=2}^{\infty }a_n z^n\) in the unit disk \(\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}\). For \(f\in \mathcal {S}\), In 1999, Ma proposed the generalized Zalcman conjecture that
with equality only for the Koebe function \(k(z) = z/(1 - z)^2\) and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of \(\lambda \) does the following inequality hold?
Clearly equality holds for the Koebe function \(k(z) = z/(1 - z)^2\) and its rotations. In this paper, we prove the inequality (0.1) for \(\lambda =3, n=2, m=3\). Further, we provide a geometric condition on extremal function maximizing (0.1) for \(\lambda =2,n=2, m=3\).
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Acknowledgements
Authors thanks Prof. Hiroshi Yanagihara and Prof. D. K. Thomas for fruitful discussions and giving constructive suggestions for improvements to this paper. The first author thanks SERB-CRG, and the second author thanks Prime Minister’s Research Fellowship (Id: 1200297) for their support.
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Allu, V., Pandey, A. On the generalized Zalcman conjecture. Annali di Matematica 203, 2665–2675 (2024). https://doi.org/10.1007/s10231-024-01461-z
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DOI: https://doi.org/10.1007/s10231-024-01461-z
Keywords
- Analytic
- Univalent
- Starlike
- Convex
- Functions
- Coefficients
- Variational method
- Zalcman conjecture
- Generalized Zalcman conjecture
- Quadratic differential