Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 5 (2010), 35 -- 47

SOME APPLICATIONS OF GENERALIZED RUSCHEWEYH DERIVATIVES INVOLVING A GENERAL FRACTIONAL DERIVATIVE OPERATOR TO A CLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS I

Waggas Galib Atshan and S. R. Kulkarni

Abstract. For certain univalent function f, we study a class of functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying

Re { (zJ1λ, μ f(z))')/((1 - γ) J1λ, μ f(z) + γ z2(J1λ, μ f(z))" )} > β.
A necessary and sufficient condition for a function to be in the class Aγλ, μ, ν(n, β) is obtained. In addition, our paper includes distortion theorem, radii of starlikeness, convexity and close-to-convexity, extreme points. Also, we get some results in this paper.

2000 Mathematics Subject Classification: 30C45.
Keywords: Distortion theorem; Radii of starlikeness; Extreme points.

Full text

References

  1. E. S. Aqlan, Some problems connected with geometric function theory, Ph.D. Thesis (2004), Pune University, Pune.

  2. P. L. Duren, Univalent Functions, Grundelheren der Mathematischen Wissenchaften 259, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. MR708494 (85j:30034). Zbl 0514.30001.

  3. S. P. Goyal and Ritu Goyal, On a class of multivalent functions defined by generalized Ruscheweyh derivatives involving a general fractional derivative operator, Journal of Indian Acad. Math. 27(2) (2005), 439-456. MR2259538 (2007d:30005). Zbl 1128.30008.

  4. S. Kanas and A. Wisniowska, Conic regions and k-uniformly convexity II, Folia Sci. Tech. Reso. 170 (1998), 65-78. MR1693661 (2000e:30017). Zbl 0995.30013.

  5. R. K. Raina and T. S. Nahar, Characterization properties for starlikeness and convexity of some subclasses of analytic functions involving a class of fractional derivative operators, Acta Math. Univ. Comen., New Ser. 69, No.1, 1-8 (2000). ISSN 0862-9544. MR1796782 (2001h:30014). Zbl 0952.30011.

  6. V. Ravichandran, N. Sreenivasagan, and H. M. Srivastava, Some inequalities associated with a linear operator defined for a class of multivalent functions, JIPAM, J. Inequal. Pure Appl. Math. 4, No. 4, Paper No. 70, 12 p., electronic only (2003). ISSN 1443-5756. MR2051571 (2004m:30022). Zbl 1054.30013.

  7. T. Rosy, K. G. Subramanian and G. Murugusundaramoorthy, Neighbourhoods and partial sums of starlike functions based on Ruscheweyh derivatives, JIPAM, J. Inequal. Pure Appl. Math. 4, No. 4, Paper No. 64, 8 p., electronic only (2003). ISSN 1443-5756. MR2051565.  Zbl 1054.30014.

  8. S. Shams and S. R. Kulkarni, Certain properties of the class of univalent functions defined by Ruscheweyh derivative, Bull. Calcutta Math. Soc. 97 (2005), 223-234. MR2191072. Zbl 1093.30012.

  9. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116. MR0369678 (51 #5910). Zbl 0311.30007.

  10. H. M. Srivastava, Distortion inequalities for analytic and univalent functions associated with certain fractional calculus and other linear operators (In Analytic and Geometric Inequalities and Applications eds. T. M. Rassias and H. M. Srivastava), Kluwar Academic Publishers, 478 (1999), 349-374. MR1785879 (2001h:30016). Zbl 0991.30007.

  11. H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Applied Mathematics and Computation, 118 (2001), 1-52. MR1805158 (2001m:26016). Zbl 1022.26012.

  12. A. Tehranchi and S. R. Kulkarni, Study of the class of univalent functions with negative coefficients defined by Ruscheweyh derivatives (II), J. Rajasthan Acad. Phy. Sci., 5(1) (2006), 105-118. MR2214020. Zbl 1138.30015.





Waggas Galib Atshan S. R. Kulkarni
Department of Mathematics, Department of Mathematics,
College of Computer Science and Mathematics, Fergusson College, Pune - 411004,
University of AL-Qadisiya, Diwaniya, Iraq. India.
e-mail: waggashnd@yahoo.com e-mail: kulkarni-ferg@yahoo.com

http://www.utgjiu.ro/math/sma