Document Type : Research Paper
Authors
1 Department of Mathematics, Faculty of Science, PNU University, P.O.BOX 19395-4697, Tehran, Iran.
2 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.
Abstract
In this paper we define a new subclass $S_{LH}(k, \gamma; \varphi)$ of log-harmonic mappings, and then basic properties such as dilations, convexity on one direction and convexity of log functions of convex- exponent product of elements of that class are discussed. Also we find sufficient conditions on $\beta$ such that $f\in S_{LH}(k, \gamma; \varphi)$ leads to $F(z)=f(z)|f(z)|^{2\beta}\in S_{LH}(k, \gamma, \varphi)$. Our results generalize the analogues of the earlier works in the combinations of harmonic functions.
Keywords
[3] Z. Abdulhadi and R.M. Ali, Univalent log-harmonic mapping in the plane, J. Abstr. Appl., Sci. Rep., 2012 (2012), pp. 1-32.
[5] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math., 9 (1984), pp. 3-25.
[7] C. Pommerenke, On starlike and close-to-convex functions, Proc. London Math. Soc., 13 (1963), pp. 290-304.
[8] W.C. Royster and M. Ziegler, Univalent functions convex in one direction, Publ. Math. Debrecen, 23 (1976), pp. 339-345.
[9] Y. Sun, Y. Jiang and Z. Wang, On the convex combinations of slanted half-plane harmonic mappings, Houston. J. Math. Anal. Appl., 6 (2015), pp. 46-50.