Galoitica: Journal of Mathematical Structures and Applications
Journal DOI
2834-5568ISSN (Online)
Volume 10 , Issue 2 , PP: 61-65, 2024 | Cite this article as | XML | Html | PDF | Full Length Article
Audy Hatim Saheb 1 *
Doi: https://doi.org/10.54216/GJMSA.0100206
In this work, we found sharp estimates for the Zalcman conjecture and second order Hankel determinant for the inverse function when it belongs to the class of starlike functions with respect to symmetric points, denoted by . These results are new.
Univalent function, regular function , Mobius transformation , close-to-convex , bi-univalent , Hankel Determinant , Zalcman conjecture.
[1] Grownwall, T., Some remarks on conformal representation, Ann. of Math., Vol.16, No.1, pp.72-76, (1914-15).
[2] Bieberbach, L., Uber. die koeffizienten. Derjenigen, Pitebzrreihen, Welcheeine Schlichte abbbiludung Des. Einheit Skreises, Vermietetn S. B. Preu. Akad. wiss., Berlin, Vol.138, pp.940-955, (1916).
[3] Alexander J. W., Functions which map the interior of the unit circle upon simple regions. Ann. Math., Vol.17, No.1, pp.12-22, (1915).
[4] MacGregor T. H., Functions whose derivative have a positive real part, Trans. Amer. Math. Soc., Vol.104, No.3, pp.532-537, (1962).
[5] Herzog, F., and Piranian, G., On the univalence of functions whose derivative has a positive real part, Proc. Amer. Math. Soc., Vol.2, pp.625-633, (1951).
[6] Goel, R. M., A class of analytic functions whose derivatives have positive real part in the unit disc, Ind. J. Math., Vol.13, pp.141-145, (1971).
[7] Goodman, A.W., Univalent functions, Mariner Publishing Comp. Inc. Tampa, Florida, Vol.I and II, (1983).
[8] Sakaguchi K., On a certain univalent mapping, J. Math. Soc. Japan, Vol.11, pp.72-75, (1959).
[9] Robertson, M. S.,On the theory of univalent functions, Ann. of Math., Vol.37, No.2, pp.374-408, (1936).
[10] Ali R. M., Lee S. K., Ravichandran V. and Supramaniam S., The Fekete-Szeg¨ o coefficient functional for transforms of analytic functions, Bull. Iran. Math. Soc., Vol.35, No.2, pp.119-142, (2009).
[11] Das R. N. and Singh P. , On subclass of schlicht mappings, Indian J. Pure Appl. Math., Vol.8, pp.864-872, (1977).
[12] Auof, M. K., On certain classes of p-valent functions, Int. J. Math. Math. Sci., Vol.9, No.1, pp.55-64, (1998).
[13] Ali, R. M., Ravichandran, V., Seenivasagan, N., Coefficient bounds for p valent finctions, Appl. Math. Comput., Vol.187, No.1, pp.35-46, (2007).
[14] Goel, R. M., and Sohi, N. S., New criteria for p-valent functions, Indian J. Pure Appl. Math., Vol.11, No.10, pp.1356-1360, (1980).
[15] Pommerenke, Ch., On the Hankel determinants of univalent functions, Math ematika, Vol.14, No.1, pp.108-112, (1967).
[16] Pommerenke, Ch., On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., Vol.41, No.s-1, pp.111-122, (1966).
[17] Dienes, P., The Taylor series, Dover., Newyork, (1957).
[18] Edrei, A., Sur les d´eterminants r´ecurrents et les singularit´es d’une fonction donn´ee par son d´eveloppement de Taylor., Compos. Math., Vol.7, No.4, pp.20-88, (1940).
[19] Hayman, W. K., On the second Hankel determinant of mean univalent func tions, Proc. London Math. Soc., Vol.18, No.3, pp.77-94, (1968).
[20] Krushkal S. L., Proof of the Zalcman conjecture for initial coefficients, Geor gian Math. J., Vol.17, pp.663-681, (2010).
[21] Krushkal S. L., Univalent functions and holomorphic motions, J. Analyse Math., 66(1995), 253–275Vol.66, pp.253-275, (1995).
[22] Ma W., Generalized Zalcman conjecture for starlike and typically real func tions, J. Math. AnaL. Appl., pp.328-339, (1999).
[23] Ma W., The Zalcman conjecture for close-to-convex functions, Proc. Amer. Math. Soc., Vol.104, pp.741-744, (1988).
[24] Pommerenke, Ch., Univalent Functions. With a Chapter on Quadratic Differ entials by Gerd Jensen. Studia Mathematic Band XXV. GmbH: Vandenhoeck and Ruprecht, (1975).
