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Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions*

  • H. M. Srivastava (a1) and Shigeyoshi Owa (a2)
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By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disk are introduced and studied systematically. The various results presented here include, for example, a number of coefficient estimates and distortion theorems for functions belonging to these subclasses, some interesting relationships between these subclasses, and a wide variety of characterization theorems involving a certain functional, some general functions of hypergeometric type, and operators of fractional calculus. Some of the coefficient estimates obtained here are fruitfully applied in the investigation of certain subclasses of analytic functions with fixed finitely many coefficients.

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      Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions*
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*

The present investigation was carried out at the University of Victoria while the second author was on study leave from Kinki University, Osaka, Japan.

This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant A-7353.

Footnotes
References
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[ 1 ] Bernardi, S. D., Convex and starlike univalent functions, Trans. Amer. Math. Soc, 135 (1969), 429446.
[ 2 ] Carlson, B. C. and Shaffer, D. B., Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737745.
[ 3 ] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, London and Toronto, 1954.
[ 4 ] Jack, I. S., Functions starlike and convex of order a, J. London Math. Soc. (2), 3 (1971), 469474.
[ 5 ] Libera, R. J., Some classes of regular univalent functions, Proc. Amer. Math. Soc, 16 (1965), 755758.
[ 6 ] Livingston, A. E., On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc, 17 (1966), 352357.
[ 7 ] MacGregor, T. H., The radius of convexity for starlike functions of order ½, Proc Amer. Math. Soc, 14 (1963), 7176.
[ 8 ] Owa, S., On the distortion theorems. I, Kyungpook Math. J., 18 (1978), 5359.
[ 9 ] Owa, S. and Srivastava, H. M., Univalent and starlike generalized hypergeometric functions, Canad. Math. J., (to appear).
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[11] Robertson, M. S., On the theory of univalent functions, Ann. of Math., 37 (1936), 374408.
[12] Ross, B., A brief history and exposition of the fundamental theory of fractional calculus, in Fractional Calculus and Its Applications (Ross, B., ed.), Springer-Verlag, Berlin, Heidelberg and New York, 1975, pp. 136.
[13] Schild, A., On starlike functions of order a, Amer. J. Math., 87 (1965), 6570.
[14] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc, 51 (1975), 109116.
[15] Srivastava, H. M. and Buschman, R. G., Convolution Integral Equations with Special Function Kernels, John Wiley and Sons, New York, London, Sydney and Toronto, 1977.
[16] Srivastava, H. M. and Kashyap, B. R. K., Special Functions in Queuing Theory and Related Stochastic Processes, Academic Press, New York and London, 1982.
[17] Srivastava, H. M. and Owa, S., An application of the fractional derivative, Math. Japon., 29 (1984), 383389.
[18] Srivastava, H. M., Owa, S. and Nishimoto, K., Some fractional differintegral equations, J. Math. Anal. Appl., 106 (1985), 360366.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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