Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912862025285632 |
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| author | Giri, Surya |
| author_facet | Giri, Surya |
| contents | Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality
$$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$
holds for functions $f(z)=z+a_2z^2 +a_3 z^3 +\cdots\in \mathcal{S}^*$, the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for $m=2$ and $n=3$, considering the class of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22558 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables Giri, Surya Complex Variables 32H02, 30C45 Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality $$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$ holds for functions $f(z)=z+a_2z^2 +a_3 z^3 +\cdots\in \mathcal{S}^*$, the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for $m=2$ and $n=3$, considering the class of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. |
| title | Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables |
| topic | Complex Variables 32H02, 30C45 |
| url | https://arxiv.org/abs/2601.22558 |