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Applications of fractional calculus to parabolic starlike and uniformly convex functions. (English) Zbl 0948.30018
Let $${\mathcal A}$$ be the class of analytic functions in the open unit disk $${\mathcal U}$$. Given $$0\leq\lambda<1$$, let $$\Omega^\lambda$$ be the operator defined on $${\mathcal A}$$ by $(\Omega^\lambda f)(z)=\Gamma(2-\lambda) z^\lambda D^\lambda_z f(z),$ where $$D^\lambda_zf$$ is the fractional derivative of $$f$$ of order $$\lambda$$. A function $$f$$ in $${\mathcal A}$$ is said to be in the class $$SP_\lambda$$ if $$\Omega^\lambda f$$ is a parabolic starlike function. In this paper several basic properties and characteristics of the class $$SP_\lambda$$ are investigated. These include subordination, inclusion, and growth theorems, class-preserving operators, Fekete-Szegő problems, and sharp estimates for the first few coefficients of the inverse function.
Reviewer: H.M.Srivastava

MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 26A33 Fractional derivatives and integrals 33C15 Confluent hypergeometric functions, Whittaker functions, $${}_1F_1$$
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References:
 [1] Duren, P.L., Univalent functions, () · Zbl 0398.30010 [2] Goodman, A.W., On uniformly convex functions, Ann. polon. math., 56, 87-92, (1991) · Zbl 0744.30010 [3] Ma, W.; Minda, D., Uniformly convex functions, Ann. polon. math., 57, 165-175, (1992) · Zbl 0760.30004 [4] Rønning, E., Uniformly convex functions and a corresponding class of starlike functions, (), 189-196 · Zbl 0805.30012 [5] Carlson, B.C.; Shaffer, D.B., Starlike and prestarlike hypergeometric functions, SIAM J. math. anal., 15, 735-745, (1984) · Zbl 0567.30009 [6] Owa, S.; Srivastava, H.M., Univalent and starlike generalized hypergeometric functions, Canad. J. math., 39, 1057-1077, (1987) · Zbl 0611.33007 [7] Owa, S., On the distortion theorems I, Kyungpook math. J., 18, 53-59, (1978) · Zbl 0401.30009 [8] Srivastava, H.M.; Owa, S., An application of the fractional derivative, Math. japon., 29, 383-389, (1984) · Zbl 0522.30011 [9] Srivastava, H.M.; Owa, S., Univalent functions, fractional calculus, and their applications, (1989), Halsted Press/John Wiley and Sons, Chichester/New York · Zbl 0683.00012 [10] Srivastava, H.M.; Mishra, A.K.; Das, M.K., A nested class of analytic functions defined by fractional calculus, Comm. appl. anal., 2, 321-332, (1998) · Zbl 0897.30003 [11] Ruscheweyh, St.; Sheil-Small, T., Hadamard products of schlicht functions and the Pólya-schoenberg conjecture, Comment. math. helv., 48, 119-135, (1973) · Zbl 0261.30015 [12] Ruscheweyh, St.; Stankiewicz, J., Subordination under convex univalent functions, Bull. Polish acad. sci. math., 33, 499-502, (1985) · Zbl 0583.30015 [13] Ling, Y.; Ding, S., A class of analytic functions defined by fractional derivation, J. math. anal. appl., 186, 504-513, (1994) · Zbl 0813.30016 [14] Goluzin, G.M., On the majorization principle in function theory, Dokl. akad. nauk SSSR, 42, 647-650, (1935), (in Russian) [15] Pommerenke, Ch., Univalent functions, (1975), Vandenhoeck and Ruprecht, Gőttingen · Zbl 0283.30034 [16] Bernardi, S.D., Convex and starlike univalent functions, Trans. amer. math. soc., 135, 429-446, (1969) · Zbl 0172.09703 [17] Nehari, Z.; Netanyahu, E., On the coefficients of meromorphic schicht functions, (), 15-23 · Zbl 0079.10004 [18] Ma, W.; Minda, D., Uniformly convex functions II, Ann. polon. math., 58, 275-285, (1993) · Zbl 0792.30008 [19] Libera, R.J.; Zlotkiewicz, E.L., Early coefficients of the inverse functions of a regular convex function, (), 225-230 · Zbl 0464.30019 [20] Ruscheweyh, St., Convolutions in geometric function theory, () · Zbl 0575.30008 [21] Yang, D.-G., The subclass of starlike functions of order λ, Chinese ann. math. ser. A, 8, 687-692, (1987) · Zbl 0656.30007
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