SOME APPLICATIONS OF FRACTIONAL CALCULUS OPERATORS TO THE ANALYTIC PART OF HARMONIC UNIVALENT FUNCTIONS

Year 2005, Volume: 34 Issue: 1, 1 - 7, 01.02.2005

Contents Of Volume 34 Mathematics B. A. Frasin B. A. Frasin

Abstract

Recently, Jahangiri [4] studied the harmonic starlike functions of order α, and he defined the class TH(α) consisting of functions f = h + g¯, where h and g are the analytic and the co-analytic part of the function f, respectively. In [3] the author introduced the class TH(α, β) of analytic functions and he proved various coefficient inequalities and growth and distortion theorems, and obtained the radius of convexity for the function h if the function f belongs to the classes TH(α) and TH(α, β). In this paper, we derive various distortion theorems for the fractional calculus and the fractional integral operator of the function h, the analytic part of the function f, if the function f belongs to the class TH(α, β).

Keywords

Harmonic, Analytic and univalent functions, Fractional calculus and fractional integral operator, 2000 AMS Classification: 30 C 45

SOME APPLICATIONS OF FRACTIONAL CALCULUS OPERATORS TO THE ANALYTIC PART OF HARMONIC UNIVALENT FUNCTIONS

Abstract

References

• Clunie, J. and Sheil-Small, T. Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 3–25, 1984.
• Erd´elyi, A., Magnus, W., Oberhettinger, F. and Tricemi, F. G. Tables of Integral Trans- forms, vol. II, (McGraw-Hill Book Co., NewYork, Toronto and London, 1954).
• Frasin, B. A. On the analytic part of harmonic univalent functions, Bull. Korain J. Math. Soc. 42 (3), 563–569, 2005.
• Jahangiri, J. M. Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235, 470–477, 1999. [5] Jahangiri, J. M., Kim, Y. C. and H. M. Srivastava, H. M. Construction of a certain class of harmonic close-to-convex functions associated with the Alexander integral transform, Integral Transform. Spec. Funct. 14, 237–242, 2003.
• Oldham, K. B. and Spanier, T. The Fractional Calculus: Theory and Applications of Differ- entiation and Integral to Arbitrary Order, (Academic Press, NewYork and London, 1974). [7] Owa, S. On the distortion theorems I, Kyungpook Math. J. 18, 53–59, 1978.
• Owa, S., Saigo, M. and Srivastava, H. M. Some characterization theorems for starlike and convex functions involving a certain fractional integral operators, J. Math. Anal. 140, 419– 426, 1989.
• Saigo, M. A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. College General Ed. Kyushu Univ. 11,135–143, 1978.
• Samko, S. G., Kilbas, A. A. and Marchev, O. I. Integrals and Derivatives of Fractional Order and Some of Their Applications, (Russian), (Nauka i Teknika, Minsk, 1987).
• Silverman, S. Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220, 283–289, 1998.
• Silverman, H. and Silvia, E. M. Subclasses of harmonic univalent functions, New Zeal. J. Math. 28, 275–284, 1999.
• Srivastava, H. M. and Buchman, R. G. Convolution Integral Equations with Special Func- tions Kernels, (John Wiely and Sons, NewYork, London, Sydney and Toronto, 1977).
• Srivastava, H. M. and Owa, S. (Eds.), Univalent functions, Fractional Calculus, and Their Applications, (Halsted Press (Ellis Horworod Limited, Chichester), John Wiely and Sons, NewYork, Chichester, Brisbane and Toronto, 1989).
• Srivastava, H. M. and Owa, S. An application of the fractional derivative, Mah. Japon. 29, 383–389, 1984. [16] Srivastava, H. M., Saigo, M. and Owa, S. A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl. 131, 412–420, 1988.