Some results on certain classes of multivalently analytic functions based on differential subordination involving a convolution structure

Mathematica Slovaca - Tập 60 - Trang 471-484 - 2010
J. K. Prajapat1, R. K. Raina2
1Department of Mathematics, Central University of Rajasthan, Kishangarh, Dist-Ajmer, India
2M.P. University of Agriculture and Technology, Udaipur, India

Tóm tắt

Making use of the familiar convolution structure of analytic functions, we introduce a general class of multivalently analytic functions and derive various useful properties and characteristics of this function class by using the techniques of differential subordination. Several other results are presented exhibiting relevant connections to some of the results obtained in earlier works.

Tài liệu tham khảo

AGHALARY, A.— JAHANGIRI, J. M.— KULKARNI, S. R.: Starlikeness and convexity conditions for class of functions defined by subordination, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), No. 2, (Article 31) 1–11 (Electronic). AOUF, M. K.— DARWISH, H. E.: Subclasses of analytic functions associated with the generalized hypergeometric function, Comput. Math. Appl. 57 (2009), 356–366. BERNARDI, S. D.: Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446. DINGGONG, Y.— LIU, J. L.: On a class of analytic functions involving Ruscheweyh derivatives, Bull. Korean Math. Soc. 39 (2002), 123–131. DZIOK, J.: Applications of the Jack Lemma, Acta Math. Hungar. 105 (2004), 93–102. DZIOK, J.— RAINA, R. K.: Families of analytic functions associated with the Wright generalized hypergeometric functions, Demonstratio Math. 37 (2004), 533–542. DZIOK, J.— RAINA, R. K.— SRIVASTAVA, H. M.: Some classes of analytic functions associated with operators on Hilbert space involving Wright’s generalized hypergeometric functions, Proc. Janggeon Math. Soc. 7 (2004), 43–55. DZIOK, J.— SRIVASTAVA, H. M.: Classes of analytic functions associated with the generalized hypergeometric functions, Appl. Math. Comput. 103 (1999), 1–13. DZIOK, J.— SRIVASTAVA, H. M.: Certain subclasses of analytic functions associated with the generalized hypergeometric functions, Integral Transform Spec. Funct. 14 (2003), 7–18. Handbook of Mathematical Functions and Formulas. Graphs and Mathematical Tables (M. Abramowitz, I. A. Stegun, eds.), Dover Publications, New York, 1971. LIU, J. L.: On subordination for certain multivalent analytic functions associated with the generalized hypergeometric function, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), No. 4, (Article 131) 1–6 (Electronic). MILLER, S. S.— MOCANU, P. T.: Differential subordinations and univalent functions, Michigan Math. J. 28 (1981), 157–171. NOOR, K. I.: On certain classes of analytic functions, JIPAM. J. Inequal. Pure and Appl. Math. 7 (2006), No. 2, (Article 49) 1–5 (Electronic). OBRADOVIC, M.: On certain inequalities for some regular functions in |z| < 1, Int. J. Math. Math. Sci. 8 (1985), 677–681. PRAJAPAT, J. K.— RAINA, R. K.: Certain results on differential subordination for some classes of multivalently analytic functions associated with a convolution structure, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 153–165. PRAJAPAT, J. K.— RAINA, R. K.: Differential subordination for certain classes of multivalently analytic functions involving convolution structure, East Asian Math. J. 25 (2009), 127–140. SHANMUGAM, T. N.— RAVICHANDRAN, V.— SIVASUBRAMANIAN, S.: Differential sandwich theorems for some subclasses of analytic functions, Austral. Math. Anal. Appl. 3 (2006), No. 1, (Article 8) 1–11 (Electronic). SRIVASTAVA, H. M.— PATEL, J.— MOHAPATRA, G. P.: Some applications of differential subordination to a general class of multivalent functions, Adv. Stud. Contemp. Math. 3 (2001), 1–15. Univalent functions, Fractional calculus, and Their Applications (H. M. Srivastava, S. Owa, eds.), Ellis Horwood Limited/Halsted Press, a division of John Wiley & Sons, Chichester/New York-Brisbane-Toronto, 1989.