## An application of fractional calculus to a new class of multivalent functions with negative coefficients.(English)Zbl 0940.30006

From the summary: Motivated by some earlier works of M.-P. Chen and H. M. Srivastava [Comput. Math. Appl. 35, No. 5, 83-91 (1998; Zbl 0921.30012)]; H. M. Srivastava and M. K. Aouf [J. Math. Anal. Appl. 171, 1-13 (1992; Zbl 0760.30006); ibid. f92, 673-688 (1995; Zbl 0831.30008)], dealing with various applications of the operators of fractional calculus in analytic function theory, the authors introduce and study rather systematically a certain subclass of analytic and $$p$$-valent functions with negative coefficients. This subclass is defined by using a familiar fractional derivative operator. Coefficient estimates, growth and distortion theorems, and many other interesting and useful properties and characteristics of this class of analytic and $$p$$-valent functions are obtained; some of these properties involve, for example, linear combinations and modified Hadamard products (or convolution) of 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.)

### Citations:

Zbl 0921.30012; Zbl 0760.30006; Zbl 0831.30008
Full Text:

### References:

 [1] Chen, M.-P.; Irmak, H.; Srivastava, H.M., A certain subclass of analytic functions involving operators of fractional calculus, Computers math. applic., 35, 5, 83-91, (1998) · Zbl 0921.30012 [2] Srivastava, H.M.; Aouf, M.K.; Srivastava, H.M.; Aouf, M.K., A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. I and II, J. math. anal. appl., J. math. anal. appl., 192, 673-688, (1995) · Zbl 0831.30008 [3] Goodman, A.W., An invitation to the study of univalent and multivalent functions, Internat. J. math. and math. sci., 2, 163-186, (1979) · Zbl 0438.30001 [4] () [5] Duren, P.L., Univalent functions, () · Zbl 0398.30010 [6] (), John Wiley and Sons, New York [7] Owa, S., On the distortion theorems. I, Kyungpook math. J., 18, 53-59, (1978) · Zbl 0401.30009 [8] Owa, S.; Srivastava, H.M., Univalent and starlike generalized hypergeometric functions, Canad. J. math., 39, 1057-1077, (1987) · Zbl 0611.33007 [9] Schild, A.; Silverman, H., Convolution of univalent functions with negative coefficients, Ann. univ. mariae Curie-skłodowska sect. A, 29, 99-107, (1975) · Zbl 0363.30018
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