Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in $$\mathbb {C}^n$$
Tóm tắt
Let
$$\widehat{\mathcal {S}}_g^{\alpha , \beta }(\mathbb {B}^n)$$
be a subclass of normalized biholomorphic mappings defined on the unit ball in
$$\mathbb {C}^n,$$
which is closely related to the starlike mappings. 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 mapping (resp. starlike mapping) are partly established also. At last, we study the Kirwan and Pell type results for the compact set of mappings which have g-parametric representation associated with a modified Roper–Suffridge extension operator, which extend some earlier related results.
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