Loewner chains and the Roper-Suffridge extension operator

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:247

Article

Graham, I ; Kohr, G ; Kohr, M

关键词: starlikeness;    spirallikeness;    convexity;    Loewner chains;    radius of starlikeness;    radius of convexity;   

DOI  :  10.1006/jmaa.2000.6843

来源: Elsevier

【 摘 要 】

Let f be a locally univalent function on the unit disc and let alpha is an element of [0, 1/2]. We consider the family of operators extending f to a holomorphic map from the unit ball B in C-n to C-n given by Phi(n, alpha)(f)(z) = (f(z(1)), z'(f'(z(1)))(alpha)), where z' = (z(2),....z(n)). When alpha = 1/2 we obtain the Roper-Suffridge extension operator. We show that if f is an element of S then Phi(n, alpha)(f) can be imbedded in a Loewner chain. Our proof shows that if f is an element of S* then Phi(n, alpha)(f) is starlike, and if f is an element of (S) over cap(beta) with \beta\ < pi/2 then Phi(n, alpha)(f) is a spirallike map of type beta. In particular we obtain a new proof that the Roper-Suffridge operator preserves starlikeness. We also obtain the radius of starlikeness of Phi(n, alpha)(S) and the radius of convexity of Phi(n, 1/2)(S). We show that if f is a normalized univalent Bloch function on U then Phi(n, alpha)(f) is a Bloch mapping on B. Finally we show that if f belongs to a class of univalent functions which satisfy growth and distortion results, then Phi(n, alpha)(f) satisfies related growth and covering results. (C) 2000 Academic Press.

【 授权许可】

Free