A new Roper-Suffridge extension operator on a Reinhardt domain
Summary: We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain \[ \Omega_{n,p_2,\dots,p_n} = \bigg\{z \in \mathbb C^n : |z_1|^2 + \sum^n_{j=2} |z_j|^{p_j} < 1\bigg\} \] given by \[ F(z) = \bigg(f(z_1) + f'(z_1) \sum^n_{j=2} a_j z^{p_j}_j\;,\; (f'(z_1))^{1/p_2} z_2\;,\; \dots, (f'(z_1))^{1/p_n} z_n\bigg)', \] where \(f\) is a normalized locally biholomorphic function on the unit disc \(\mathbb D\), \(p_j\) are positive integers, \(a_j\) are complex constants, and \(j = 2, \dots, n\). Some conditions for \(a_j\) are found under which the operator preserves almost starlike mappings of order \(\alpha\) and starlike mappings of order \(\alpha\), respectively. In particular, our results reduce to many well-known results when all \(a_j = 0\).
scientific article; zbMATH DE number 5997832
scientific article