## A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients.(English)Zbl 0760.30006

Making use of a certain operator of fractional derivatives, a new subclass $${\mathcal J}_ p(\alpha,\beta,\lambda)$$ of analytic and $$p$$-valent functions with negative coefficients is introduced and studied here rather systematically. Coefficient estimates, distortion theorems, and various other interesting and useful properties of this class of functions are given; some of these properties involve, for example, linear combinations and modified Hadamard products of several functions belonging to the class introduced here.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions of one complex variable
Full Text:

### References:

 [1] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., () [2] Gupta, V.P.; Jain, P.K., Certain classes of univalent functions with negative coefficients. II, Bull. austral. math. soc., 15, 467-473, (1976) · Zbl 0335.30010 [3] Oldham, K.B.; Spanier, J., The fractional calculus: theory and applications of differentiation and integration to arbitrary order, (1974), Academic Press New York/London · Zbl 0292.26011 [4] Owa, S., On the distortion theorems. I, Kyungpook math. J., 18, 53-59, (1978) · Zbl 0401.30009 [5] Owa, S., On certain subclasses of analytic p-valent functions, J. Korean math. soc., 20, 41-58, (1983) · Zbl 0532.30016 [6] Owa, S.; Saigo, M.; Srivastava, H.M., Some characterization theorems for starlike and convex functions involving a certain fractional integral operator, J. math. anal. appl., 140, 419-426, (1989) · Zbl 0668.30013 [7] Saigo, M., A remark on integral operators involving the Gauss hypergeometric functions, Math. rep. college general ed. kyushu univ., 11, 135-143, (1978) [8] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Integrals and derivatives of fractional order and some of their applications, (1987), Nauka i Tekhnika Minsk, [In Russian] · Zbl 0617.26004 [9] Srivastava, H.M.; Buschman, R.G., Convolution integral equations with special function kernels, (1977), Wiley New York/London/Sydney/Toronto · Zbl 0346.45010 [10] Srivastava, H.M.; Owa, S., An application of the fractional derivative, Math. japon., 29, 383-389, (1984) · Zbl 0522.30011 [11] Srivastava, H.M.; Owa, S., A new class of analytic functions with negative coefficients, Comment. math. univ. st. paul., 35, 175-188, (1986) · Zbl 0588.30012 [12] (), (Ellis Horwood Limited, Chichester), Wiley, New York/Chichester/Brisbane/Toronto
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.