# The Distortion in Conformal Mapping in Hyperbolic Geometry

## Keywords:

Distortion, Conformal maps, Unit disc, Sharp two-point, hyperbolic geometry## Abstract

Nehari-type inequalities for normalized univalent functions are combined with elementary monotonicity arguments to give quick and simple proofs for numerous sharp two-point distortion theorems for conformal maps. These inequalities are used to prove two theorems of sharp two-point distortion theorems for conformal maps from the unit disk into the unit disk equipped with hyperbolic geometry.

## References

Blatter, CH.(1978), ″Ein Verzerrungssatz fur Schichte Funktionen,″ Commet. Math. Helv. 53,651-659 .

El-Gawi,S and shok, A.[2009],″ The Distortion in Conformal Mapping in Euclidean Geometry, ″ Journal of Basic and Applied Sciences, Volume 19, Issue no 2.

Jenkins, J. A. (1998), ″On Weighted Distortion in Conformal Mapping II,″ Bull. London Math. Soc 30, 151-158.

Jenkins, J. A. (2001), ″On Two-point Distortion Theorems for Bounded Univalent Regular Function,″ Kodai Math. J. 24:3, 329-338.

Kim, S. and Minda, D. (1994), ″ Two-point Distortion Theorems for Univalent Function, ″ Pacific J. Math. 163, 137-157.

Ma, W. and Minda, D. (1997), ″ Two-point Distortion Theorems for Bounded Univalent Functions,″ Ann. Acad. Sci. Fenn. Math. 22, 425-444.

Ma, W. and Minda, D. (1999), ″ Two-point Distortion for Univalent Function,″ J. Comput. Appl. Math. 105, 385-392.

Ma, W. and Minda, D. (2000), ″ Two-point Distortion Theorems for Spherically Convex Functions,″ Rocky Mount. J. Math. 30:2, 663-687..

Marshall, A. W., olkin I. and Proschan, F. (1967), ″ Monotonicity of Ratios of Means and other Applications of Majorization, ″ In: Inegualities, edited by O. Shisha. Acad. Press. 177- 190.

Nehari, Z.(1953), ″ Some Inequalities in the Theory of functions, ″ Trans. Amer. Math. Soc. 75, 256-286.

Pick, G. (1917), ″ Uber die Konforme Abbildung eines Kreises auf ein Schlichtes und Zugleich beschranktes Gebiet, ″ S. B. Kaiserl. Akad. Wiss. Wien, Math. Naturwiss, K1, 126, 247-263.

Robinson, R. M. (1942), ″ Bounded Univalent Functions, ″ Trans. Amer. Math. Soc. 52, 426-449.

Roth, O. (2002), ″ A Distortion Theorem for Bounded Univalent Functions, ″ Ann. Acam. Sci. Fenn. Math. 27, 257-272.

Roth, O. (2003), ″ Distortion Theorems for Bounded Univalent Function, ″ Analysis 23, 347- 369.