A generalization of univalent functions with bounded boundary rotation
HTML articles powered by AMS MathViewer
- by Edward J. Moulis
- Trans. Amer. Math. Soc. 174 (1972), 369-381
- DOI: https://doi.org/10.1090/S0002-9947-1972-0320296-1
- PDF | Request permission
Abstract:
This paper introduces a class of functions which generalizes both those functions $f(z)$ with bounded boundary rotation and those functions for which $zf’(z)$ is a-spirallike. A simple variational formula for this class is derived and used to determine sufficient conditions for the univalency of functions there in. Various representations for these functions are given, and these are used to derive another condition for univalence; this one is the best known so far in the subclass consisting of functions $f(z)$ for which $zf’(z)$ is a-spirallike. Bounds on the modulus of the Schwarzian derivative are also derived; these are sharp in the subclass of functions having bounded boundary rotation.References
- Jochen Becker, Über Subordinationsketten und quasikonform fortsetzbare schlichte Funktionen, Technische Universität Berlin, Berlin, 1970. Von der Fakultät für Allgemeine Ingenieurwissenschaften der Technischen Universität Berlin zur Verleihung des akademischen Grades Doktor der Naturwissenschaften genehmigte Dissertation. MR 0437751
- D. A. Brannan, On functions of bounded boundary rotation. I, Proc. Edinburgh Math. Soc. (2) 16 (1968/69), 339–347. MR 264045, DOI 10.1017/S001309150001302X H. B. Coonce, A variational formula for functions of bounded boundary rotation, Ph. D. Dissertation, University of Delaware, Newark, Del., 1969, 34 pp.
- Olli Lehto, On the distortion of conformal mappings with bounded boundary rotation, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1952 (1952), no. 124, 14. MR 53241 K. Lowner, Untersuchen über die Verzerrung bei konformen Abbildungen Einheitskreises $|z| < 1$, die durch Funktionen mit nicht verschwindender Ableitung geleifiet werden, Ber. Konigl. Sachs. Ges. Wiss. Leipzig 69 (1917), 89-106.
- Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. MR 29999, DOI 10.1090/S0002-9904-1949-09241-8 V. Paatero, Über die konforme Abbildung von Gebieten deren Rander von beschrankter Drehung sind, Ann. Acad. Sci. Fenn. Ser. A 33 (1931), 77 pp. —, Über Gebiete von beschrankter Randdrehung, Ann. Acad. Sci. Fenn. Ser. A 37 (1933), 20 pp.
- M. S. Robertson, Coefficients of functions with bounded boundary rotation, Canadian J. Math. 21 (1969), 1477–1482. MR 255798, DOI 10.4153/CJM-1969-161-2
- M. S. Robertson, Univalent functions $f(z)$ for which $zf^{\prime } (z)$ is spirallike, Michigan Math. J. 16 (1969), 97–101. MR 244471
- Malcolm S. Robertson, Variational formulae for several classes of analytic functions, Math. Z. 118 (1970), 311–319. MR 0281900, DOI 10.1007/BF01109867 M. R. Ziegler, A class of regular functions related to univalent functions, Ph. D. Dissertation, University of Delaware, Newark, Del., 1970, 77 pp.
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 174 (1972), 369-381
- MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9947-1972-0320296-1
- MathSciNet review: 0320296