Polylogarithms in the Theory of Univalent Functions

Results in Mathematics

Volumn 30, Issue 1-2, 1996, Pages 136-150

  Ponnusamy, S ^a   Sabapathy, S ^b  

^a UNIVERSITY OF HELSINKI   (Finland)

^b AA Government Arts College   (India)

Author keywords

Polylogarithm; starlike and generalized hypergeometric functions; univalent

Indexed keywords

EID: 2042467817     PISSN: 14226383     EISSN: 14209012     Source Type: Journal    
DOI: 10.1007/BF03322186     Document Type: Article

References (27)

1. R. Ali and V. Singh, Convexity and starlikeness of functions defined by a class of integral operators, Complex Variables: Theory Appl. 26(1995), 299–309.
2. G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc, 347(1995), 1713–1723.
3. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Conformai invariants, inequalities, and quasiconformal maps, Book Manuscript.
4. K. Aomoto, Hypergeometric functions-past, present, and possible future (Japanese) Sugaku 45(1993), 208–220.
5. H. Bateman, (Ed. by A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi), Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
6. B. C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, Berlin — Heidelberg — New York, 1989.
7. W. Bühring, Generalized hypergeometric functions at unit argument, Proc. Amer. Math. Soc, 114(1992), 145–153.
8. P. L. Duren, Univalent functions, Springer-Verlag, 1983.
9. L. Fejér, Untersuchungen über Potenzreihen mit mehrfach monotoner Koeffizientenfolge, Ada Litterarum ac Scientiarum 8(1936), 89–115.
10. R. Fournier and St. Ruscheweyh, On two extremal problems related to univalent functions, Rocky Mountain J. Math. 24(2)(1994), 529–538.
11. C. I. Gerhardt (Editor), G. W. Leibniz, Mathematische Schriften III/1, pp. 336–339, Georg Olms Verlag, Hildesheim and New York, 1971.
12. A. B. Goncharov, Polylogarithms in arithmetic and geometry, Proceedings of the International Congress of Mathematicians, Zürich, Switzerland (1994), 374-387.
13. E. R. Hansen, A table of series and products, Prentice Hall, Inc., Englewood Cliffs, N.J., 1975.
14. ELerch1887Note sur la fonction1119241554747JFM 19.0438.01
15. L. Lewin, Polylogarithms and associated functions, Elsevier North-Holland, New York and Oxford, 1981.
16. L. Lewin, editor, Structural properties of polylogarithms, Math, surveys and monographs Vol. 37, Amer. Math. Soc, Providence, RI, 1991.
17. J. L. Lewis, Convexity of certain series, J. London Math. Soc. (3)27(1983), 435–446.
18. S. Ozaki, On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daigaku A 2(1935), 167–188.
19. S. Ponnusamy, Differential subordination and starlike functions, Complex Variables: Theory and Appln. 19(1992), 185–194. au[20]_S. Ponnusamy, Neighborhoods and Carathéodory functions, J. Analysis, to appear.
20. S. Ponnusamy, The Hardy spaces of hypergeometric functions, Complex Variables: Theory and Appln. 29 (1996), 83–96.
21. S. Ponnusamy, Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane, Prepxint 92, 1995, Department of Mathematics, University of Helsinki, 28 pp; Also Rocky Mountain J. of Mathematics, to appear.
22. S. Ponnusamy and S. Sabapathy, Geometric properties of generalized hypergeometric functions, Preprint 102, 1996, Department of Mathematics, University of Helsinki, 22 pp.
23. S. Ponnusamy and M. Vuorinen, Asymptotic expansions and inequalities for hypergeometric functions, Preprint 70, 1995, Department of Mathematics, University of Helsinki, 22 pp.
24. S. Ponnusamy and M. Vuorinen, Univalence and convexity properties of Gaussian hypergeometric functions, Preprint 82, 1995, Department of Mathematics, University of Helsinki, 34pp.
25. St. Ruscheweyh, Convolutions in geometric function theory, Séminaire de Mathématiques supérieures, NATO Advanced Study Institute, Les Presses de l’Université de Montréal, Montréal, 1982.
26. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. Cambridge Univ. Press, 1958.
27. D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-Theory of fields, Arithmetic Algebraic Geometry (G.v.d. Geer, F. Oort, and J. Steenbrink, eds.), Prog. Math., Vol. 89, Birkhäuser, Basel and Boston (1991), 391–430.