Meromorphic functions with small Schwarzian derivative
DOI:
https://doi.org/10.24193/subbmath.2018.3.07Keywords:
Meromorphic functions, convex functions, meromorphically convex functions, close-to-convex functions, starlike functions, Schwarzian derivative.Abstract
We consider the family of all meromorphic functions f of the form f (z) = 1 + b + b z + b z2 + · · · z 0 1 2 analytic and locally univalent in the puncture disk D0 := {z ∈ C : 0 < |z| < 1}. Our first objective in this paper is to find a sufficient condition for f to be meromorphically convex of order α, 0 ≤ α < 1, in terms of the fact that the absolute value of the well-known Schwarzian derivative Sf (z) of f is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions g of the form g(z) = z + a2z2 + a3z3 + · · · analytic and locally univalent in the open unit disk D := {z ∈ C : |z| < 1}, and show that g is belonging to a family of functions convex in one direction if |Sg (z)| is bounded above by a small positive constant depending on the second coefficient a2. In particular, we show that such functions g are also contained in the starlike and close-to-convex family.
Mathematics Subject Classification (2010): 30D30, 30C45, 30C55, 34M05.
References
Bharanedhar, S.V., Ponnusamy, S., Uniform close-to-convexity radius of sections of functions in the close-to-convex family, J. Ramanujan Math. Soc., 29(2014), no. 3, 243–251.
Chiang, Y.M., Properties of analytic functions with small Schwarzian derivative, Proc. Amer. Math. Soc., 24(1994), 107–118.
Duren, P.L., Univalent functions, Springer-Verlag, New York, 1983.
Gabriel, R.F., The Schwarzian derivative and convex functions, Proc. Amer. Math. Soc., 6(1955), 58–66.
Goodman, A.W., Univalent functions, Vol. 1-2, Mariner, 1983.
Hille, E., Remarks on a paper by Zeev Nehari, Proc. Amer. Math. Soc., 55(1949), 552– 553.
Kim, J.A., Sugawa, T., Geometric Properties of functions with small Schwarzian derivative, Preprint.
Lehto, O., Univalent functions and Teichmu¨ller spaces, Springer-Verlag, New York, 1987.
Li, L., Ponnusamy, S., On the generalized Zalcman functional λa2 − a , 2n−1 in the close to-convex family Proc. Amer. Math. Soc., 145(2017), no. 2, 833–846.
Miller, S.S., Mocanu, P.T., Differential subordinations: theory and applications, Dekker, New York, 2000.
Muhanna, Y.A., Li, L., Ponnusamy, S., Extremal problems on the class of convex functions of order −1/2, Arch. Math. (Basel), 103(2014), no. 6, 461–471.
Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc., 55(1949), 545–551.
Nehari, Z., Some criteria of univalence, Proc. Amer. Math. Soc., 5(1954), 700–704. [14] Nehari, Z., Univalence criteria depending on the Schwarzian derivative, Illinois J. Math., 23(1979), 345–351.
Nunokawa, M., On meromorphically convex and starlike functions, Su¯rikaisekikenkyu¯sho K¯okyu¯roku, 1164(2000), 57–62.
Pokornyi, V.V., On some sufficient conditions for univalence, (Russian), Dokl. Akad. Nauk SSSR, 79(1951), 743–746.
Pommerenke, Ch., Univalent functions, Vandenhoeck and Ruprecht, G¨ottingen, 1975. [18] Ponnusamy, S., Rajasekaran, S., New sufficient condition for starlike and univalent functions, Soochow J. Math., 21(1995), 193–201.
Ponnusamy, S., Sahoo, S.K., Norm estimates for convolution transforms of certain classes of analytic functions, J. Math. Anal. Appl., 342(2008), 171–180.
Ponnusamy, S., Sahoo, S.K., Yanagihara, H., Radius of convexity of partial sums of functions in the close-to-convex family, Nonlinear Anal., 95(2014), 219–228.
Shah, G.M., On Holomorphic functions convex in one direction, J. Indian Math. Soc., 37(1973), 257–276.
Tichmarsh, E.C., The theory of functions, 2nd ed., Oxford University Press, 1939.
Umezawa, T., Analytic functions convex in one direction, J. Math. Soc. Japan, 4(1952), 195–202.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
