Univalence criteria for locally univalent analytic functions

Zhenyong Hu; Jinhua Fan; Xiaoyuan Wang

doi:10.37863/umzh.v75i7.7222

Zhenyong Hu School of Mathematics and Statistics, Nanjing University of Science and Technology, China
Jinhua Fan School of Mathematics and Statistics, Nanjing University of Science and Technology, China
Xiaoyuan Wang School of Mathematics and Statistics, Nanjing University of Science and Technology, China

DOI: https://doi.org/10.37863/umzh.v75i7.7222

Keywords: locally univalent analytic functions, John constant, univalence criterion.

Abstract

UDC 517.5

Suppose that $p(z)=1+z\phi''(z)/\phi'(z),$ where $\phi(z)$ is a locally univalent analytic function in the unit disk $\mathbf{D}$ with $\phi(0)=\phi'(1)-1=0.$ We establish the lower and upper bounds for the best constants $\sigma_{0}$ and $\sigma_{1}$ such that $e^{-\sigma_{0}/2}<|p(z)|<e^{\sigma_{0}/2}$ and $|p(w)/p(z)|<e^{\sigma_{1}}$ for $z,w\in\mathbf{D},$ respectively, imply the univalence of $\phi(z)$ in $\mathbf{D}.$

References

D. Aharonov, U. Elias, Univalence criteria depending on parameters, Anal. and Math. Phys., 4, 23–34 (2014).

J. Becker, Lownersche Differentialgleichung und quasikonform fortsetzbare schlichet Functionen, J. reine und angew. Math., 255, 23–43 (1972).

J. Brown, Quasiconformal extensions for some geometric subclasses of univalent functions, Int. J. Math. and Math. Sci., 7, 187–195 (1984).

J. Gevirtz, An upper bound for the John constant, Proc. Amer. Math. Soc., 83, 476–478 (1981).

J. Gevirtz, On extremal functions for John constants, J. London Math. Soc., 39, 285–298 (1989).

F. John, On quasi-isometric mappings, II, Comm. Pure and Appl. Math., 22, 265–278 (1969).

I. Hotta, Explicit quasiconformal extensions and Löwner chains, Proc. Japan Acad. Ser. A Math. Sci., 85, 108–111 (2009).

J. A. Hummel, The Grunsky coefficients of a schlicht function, Proc. Amer. Math. Soc., 15, 142–150 (1964).

Y. C. Kim, T. Sugawa, Univalence criteria and analogues of the John constant, Bull. Aust. Math. Soc., 88, 423–434 (2013).

O. Lehto, Univalent functions and Teichmüller space, Grad. Texts in Math., 109, Springer-Verlag, New York (1987).

Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc., 55, 545–551 (1949).

Z. Nehari, Some criteria of univalence, Proc. Amer. Math. Soc., 5, 700–704 (1954).

Z. Nehari, Univalence criteria depending on the Schwarzian derivative, Illinois J. Math., 23, 345–351 (1979).

T. Nikola, J. Biljana, P. Boljan, On existence of sharp univalence criterion using the Schwarzian derivative, C. R. Acad. Bulgare Sci., 68, 569–576 (2015).

K. Padmanabhan, S. Kumar, On a class of subordination chains of univalent function, J. Math. Phys., 25, 361–368 (1991).

C. Pommerenke, Univalent functions, Vandenhoeck Ruprecht, Göttingen (1975).

S. Yamashita, On the John constant, Math. Z., 161, 185–188 (1978).