推广的Roper-Suffridge算子与Loewner链

将Roper-Suffridge箅子在C~n中单位球B~n上做了进一步推广,并考察推广后的算子何时能保持双全纯映照子族的性质.利用k阶零点及双全纯映照子族的增长定理,重点研究了推广后的算子在B~n上保持α次β型螺形映照及强β型螺形映照的性质,并由调和函数的最小值原理及具有正实部函数的性质,揭示了推广后的算子能够嵌入Loewner链,从而得到推广后的算子在B~n上保持α次殆β型螺形映照的性质.

The Generalized Roper-Suffridge Operator and Loewner Chains

The authors extend the Roper-Suffridge operator to the unit ball $B^n$ in $\mathbb{C}^n$, and seek conditions under which the extended operator preserves the properties of subclasses of biholomorphic mappings. By the zero of order $k$ and the growth theorems for subclasses of biholomorphic mappings, it is primarily studied that the extended operator preserves spirallikeness of order $\alpha$ and type $\beta$, strong spirallikeness of type $\beta$ on $B^{n}$. By the Minimum Principle for harmonic functions and the property of functions which have positive real parts, the extended operator can be embedded into a Loewner chain, and thus the extended operator preserves almost spirallikeness of type $\beta$ and order $\alpha$ on $B^{n}$.