# zbMATH — the first resource for mathematics

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, $${}_1F_1$$
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.