Logharmonic mappings with typically real analytic components
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This paper treats the class of normalized logharmonic mappings $f(z)=zh(z)\overline{g(z)}$ in the unit disk satisfying $\varphi(z)=zh(z)g(z)$ is analytically typically real. Every such mapping $f$ admits an integral representation in terms of its second dilatation function and a function of positive real part with real coefficients. The radius of starlikeness and an upper estimate for arclength are obtained. Additionally, it is shown that $f$ maps the unit disk into a domain symmetric with respect to the real axis when its second dilatation has real coefficients.
Abstract
This paper treats the class of normalized logharmonic mappings $f(z)=zh(z)\overline{g(z)}$ in the unit disk satisfying $\varphi(z)=zh(z)g(z)$ is analytically typically real. Every such mapping $f$ admits an integral representation in terms of its second dilatation function and a function of positive real part with real coefficients. The radius of starlikeness and an upper estimate for arclength are obtained. Additionally, it is shown that $f$ maps the unit disk into a domain symmetric with respect to the real axis when its second dilatation has real coefficients.
참고문헌 (21)
- Z. Abdulhadi, "Close-to-starlike logharmonic mappings", Internat. J. Math. Math. Sci, 19(3), pp.563~574 (1996)
- Z. Abdulhadi, "Typically real logharmonic mappings", Int. J. Math. Math. Sci, 31(1), pp.1~9 (2002)
- Z. AbdulHadi, "On the convex-exponent product of logharmonic mappings", J. Inequal. Appl, 2014, pp.485~ (2014)
- Z. Abdulhadi, "Univalent logharmonic mappings in the plane", Abstr. Appl. Anal, 2012, pp.32~ (2012)
- Z. Abdulhadi, "On rotationally starlike logharmonic mappings", Math. Nachr, 288(7), pp.723~729 (2015)
- Z. Abdulhadi, "Univalent functions in H · H(D)", Trans. Amer. Math. Soc, 305(2), pp.841~849 (1988)
- Z. Abdulhadi, "Univalent functions, fractional calculus, and their applica-tions", pp.13~28 (1989), Horwood
- Z. Abdulhadi, "Starlike log-harmonic mappings of order α", JIPAM. J. Inequal. Pure Appl. Math, 7(4), pp.6~ (2006)
- Y. Abu-Muhanna, "The boundary behaviour of harmonic univalent maps", Pacific J. Math, 141(1), pp.1~20 (1990)
- J. Clunie, "Harmonic univalent functions", Ann. Acad. Sci. Fenn. Ser. A I Math, 9, pp.3~25 (1984)
- P. L. Duren, "Univalent Functions" (1983), Springer-Verlag
- P. Duren, "A variational method for harmonic mappings onto convex regions", Complex Variables Theory Appl, 9(2-3), pp.153~168 (1987)
- P. Duren, "Linear extremal problems for harmonic mappings of the disk", Proc. Amer. Math. Soc, 106(4), pp.967~973 (1989)
- A. W. Goodman, "Univalent Functions. Vol. I" (1983), Mariner Publishing Co., Inc.
- W. Hengartner, "On the boundary behavior of orientation-preserving harmonic mappings", Complex Variables Theory Appl, 5(2-4), pp.197~208 (1986)
- W. Hengartner, "Harmonic mappings with given dilatation", J. London Math. Soc. (2), 33(3), pp.473~483 (1986)
- S. H. Jun, "Univalent harmonic mappings on Δ =｛z :|z| > 1｝", Proc. Amer. Math. Soc, 119(1), pp.109~114 (1993)
- W. E. Kirwan, "Extremal problems for the typically real functions", Amer. J. Math, 88, pp.942~954 (1966)
- J. C. C. Nitsche, "Lectures on Minimal Surfaces. Vol. 1" (1989), Cambridge University Press
- R. Osserman, "A Survey of Minimal Surfaces" (1986), Dover Publications, Inc.
- W. Rogosinski, "Uber positive harmonische Entwicklungen und typisch-reelle Potenzreihen", Math. Z, 35(1), pp.93~121 (1932)