Abstract
Let \(\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)\) be a subclass of normalized biholomorphic map**s defined on the unit ball in \(\mathbb {C}^n,\) which is closely related to the starlike map**s. Firstly, we obtain the growth theorem for \(\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)\). Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for this subclass, and the distortion theorems with g-starlike map** (resp. starlike map**) are partly established also. At last, we study the Kirwan and Pell type results for the compact set of map**s which have g-parametric representation associated with a modified Roper–Suffridge extension operator, which extend some earlier related results.
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The project is supported by the National Natural Science Foundation of China (No. 11671306).
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Tu, Z., ** and Extremal Problem with Parametric Representation in \(\mathbb {C}^n\). Complex Anal. Oper. Theory 13, 2747–2769 (2019). https://doi.org/10.1007/s11785-018-00881-z
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DOI: https://doi.org/10.1007/s11785-018-00881-z