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COEFFICIENT ESTIMATES FOR SOME CLASSES OF FUNCTIONS ASSOCIATED WITH $q$-FUNCTION THEORY

Part of: Classical measure theory Geometric function theory Elementary classical functions

Published online by Cambridge University Press:  06 March 2017

SARITA AGRAWAL
SARITA AGRAWAL*
Affiliation:
Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore 453 552, India email saritamath44@gmail.com
*
✉saritamath44@gmail.com
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Abstract

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For every $q\in (0,1)$, we obtain the Herglotz representation theorem and discuss the Bieberbach problem for the class of $q$-convex functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$. In addition, we consider the Fekete–Szegö problem and the Hankel determinant problem for the class of $q$-starlike functions, leading to two conjectures for the class of $q$-starlike functions of order $\unicode[STIX]{x1D6FC}$ with $0\leq \unicode[STIX]{x1D6FC}<1$.

Keywords

q-starlike function of order 𝛼q-convex functionHerglotz representationBieberbach’s conjectureFekete–Szegö problemHankel determinant

MSC classification

Primary: 30C50: Coefficient problems for univalent and multivalent functions
Secondary: 28A25: Integration with respect to measures and other set functions 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 33B10: Exponential and trigonometric functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 446 - 456
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Agrawal, S. and Sahoo, S. K., ‘A generalization of starlike functions of order alpha’, Hokkaido Math. J. 46 (2017), 15–27.CrossRefGoogle Scholar
Baricz, A. and Swaminathan, A., ‘Mapping properties of basic hypergeometric functions’, J. Class. Anal. 5(2) (2014), 115–128.CrossRefGoogle Scholar
de Branges, L., ‘A proof of the Bieberbach conjecture’, Acta Math. 154(1–2) (1985), 137–152.CrossRefGoogle Scholar
Duren, P. L., Univalent Functions (Springer, New York, 1983).Google Scholar
Fekete, M. and Szegö, G., ‘Eine Bemerkung über ungerade schlichte Funktionen’, J. Lond. Math. Soc. (2) 8 (1933), 85–89.CrossRefGoogle Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, 35 (Cambridge University Press, Cambridge, 1990).Google Scholar
Goodman, A. W., Univalent Functions, Vol. 1 (Mariner, Tampa, FL, 1983).Google Scholar
Ismail, M. E. H., Merkes, E. and Styer, D., ‘A generalization of starlike functions’, Complex Variables 14 (1990), 77–84.Google Scholar
Jackson, F. H., ‘On q-definite integrals’, Quart. J. Pure Appl. Math. 41 (1910), 193–203.Google Scholar
Janteng, A., ‘Hankel determinant for starlike and convex functions’, Int. J. Math. Anal. 13(1) (2007), 619–625.Google Scholar
Keogh, F. R. and Merkes, E. P., ‘A coefficient inequality for certain classes of analytic functions’, Proc. Amer. Math. Soc. 20 (1969), 8–12.CrossRefGoogle Scholar
Koepf, W., ‘On the Fekete–Szegö problem for close-to-convex functions’, Proc. Amer. Math. Soc. 101 (1987), 89–95.Google Scholar
Koepf, W., ‘On the Fekete–Szegö problem for close-to-convex functions II’, Arch. Math. 49 (1987), 420–433.CrossRefGoogle Scholar
Libera, R. J. and Zlotkiewicz, E. J., ‘Coefficient bounds for the inverse of a function with derivative in O’, Proc. Amer. Math. Soc. 87(2) (1983), 25–257.Google Scholar
London, R. R., ‘Fekete–Szegö inequalities for close-to convex functions’, Proc. Amer. Math. Soc. 117 (1993), 947–950.Google Scholar
Ma, W. and Minda, D., ‘A unified treatment of some special classes of univalent functions’, in: Proceedings of the Conference on Complex Analysis (International Press, Tianjin, 1992), 157–169.Google Scholar
Pfluger, A., ‘The Fekete–Szegö inequality by a variational method’, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 447–454.CrossRefGoogle Scholar
Pfluger, A., ‘The Fekete–Szegö inequality for complex parameters’, Complex Var. Theory Appl. 7(1–3) (1986), 149–160.Google Scholar
Sahoo, S. K. and Sharma, N. L., ‘On a generalization of close-to-convex functions’, Ann. Polon. Math. 113(1) (2015), 93–108.CrossRefGoogle Scholar
Thomae, J., ‘Beiträge zur Theorie der durch die Heinesche Reihe: … darstellbaren Funktionen’, J. reine angew. Math. 70 (1869), 258–281.Google Scholar