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On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

Published online by Cambridge University Press:  20 November 2018

Alan D. Gluchoff
Alan D. Gluchoff*
Affiliation:
Villanova University Villanova, Pennsylvania 19085 U.S.A.
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Abstract

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The purpose of this paper is to prove some facts about integral means of (d2/dz2)(log[f(z)/z])—or equivalently f″/f, for f in a class of starlike mappings of a "singular" nature. In particular it is noted that the Koebe function is not extremal for the Hardy means Mp(r,f″/f) for functions in this class.

Keywords

30C45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 1 , 01 March 1993 , pp. 38 - 44
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Ahern, P., The Mean Modulus and the Derivative of an Inner Function, Indiana University Mathematics Journal (2) 28(1979), 311–347.Google Scholar
2. Ahern, P., The Poisson Integral of a Singular Measure, Canad. J. Math. XXXV(1983), 735–749.Google Scholar
3. Ahern, P. and Clark, D., On Inner Functions With Hp Derivative, Michigan Math J. 21(1974), 115–127.Google Scholar
4. Caughren, J. G. and Shields, A., Singular Inner Factors of Analytic Functions, Michigan Math. J. 16(1969), 409–410.Google Scholar
5. Cullen, M. R., Derivatives of Singular Inner Functions, Michigan Math J. 18(1971), 283–287.Google Scholar
6. Duren, P. L., Theory of Hp Spaces, Academic Press, New York, 1970.Google Scholar
7. Duren, P. L., Univalent Functions, Springer-Verlag, New York and Heidelberg, 1983.Google Scholar
8. Duren, P. L. and Y Leung, J., Logarithmic Coefficients of Univalent Functions, Journal d'Analyse Mathématique 36(1979), 36–43.Google Scholar
9. Elhosh, M. M., On Integral Means of Derivatives of Univalent Functions, Bull. Korean Math. Soc. (1) 24(1987), 13–17.Google Scholar
10. Feng, J. and MacGregor, T. H., Estimates on Integral Means of the Derivatives of Univalent Functions, Notices Amer. Math. Soc. 21 August (1974), A-492.Google Scholar
11. Garnett, J., Bounded Analytic Functions, Academic Press, New York, 1981.Google Scholar
12. Keogh, F. R., Conformai Mappings of Bounded Star-Shaped Domains, Proc. London Math Society (3) 9(1959), 481–491.Google Scholar
13. MacGregor, T. H., Applications of Extreme Point Theory To Univalent Functions, Michigan Math J. (4) 19(1972), 361–376.Google Scholar
14. Pommerenke, C., On Starlike and Convex Functions, J. London Math. Soc. 37(1962), 209–224.Google Scholar
15. Sheil-Small, T., Starlike Univalent Functions, Proc. London Math. Soc. (3) 21(1970), 577–613.Google Scholar