Classes of analytic functions with fractional powers defined by means of a certain linear operator

Motivated by the success of the familiar Dziok–Srivastava convolution operator, we introduce here a closely-related linear operator for analytic functions with fractional powers. By means of this linear operator, we then define and investigate a class of analytic functions. Finally, we determine certain conditions under which the partial sums of the linear operator of bounded turning are also of bounded turning. We also illustrate an application of a fractional integral operator.

[1]  Hari M. Srivastava,et al.  Classes of multivalent analytic functions involving the Dziok-Srivastava operator , 2007, Comput. Math. Appl..

[2]  B. C. Carlson,et al.  Starlike and Prestarlike Hypergeometric Functions , 1984 .

[3]  G. Murugusundaramoorthy,et al.  Starlike and convex functions of complex order involving the Dziok-Srivastava operator , 2007 .

[4]  M. Darus,et al.  Coefficient inequalities for a new class of univalent functions , 2008 .

[5]  Kambiz Farahmand,et al.  Partial sums of functions of bounded turning , 2004, Int. J. Math. Math. Sci..

[6]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[7]  Hassoon S. Al-Amiri,et al.  On the radius of univalence of certain analytic functions , 1973 .

[8]  S. D. Bernardi,et al.  Convex and starlike univalent functions , 1969 .

[9]  N. Ayırtman,et al.  Univalent Functions , 1965, Series and Products in the Development of Mathematics.

[10]  M. Darus,et al.  Differential subordination results for new classes of the family ε(Φ, ψ) , 2009 .

[11]  Hari M. Srivastava,et al.  Univalent Functions, Fractional Calculus, and Their Applications , 1990, The Mathematical Gazette.

[12]  M. Darus,et al.  On Cesáro means for Fox-Wright functions , 2008 .

[13]  Maslina Darus,et al.  Subordination and superordination for univalent solutions for fractional differential equations , 2008 .

[14]  T. Macgregor.  Functions whose derivative has a positive real part , 1962 .

[15]  Hari M. Srivastava,et al.  Univalent and Starlike Generalized Hypergeometric Functions , 1987, Canadian Journal of Mathematics.

[16]  Pran Nath Chichra.  New subclasses of the class of close-to-convex functions , 1977 .

[17]  H. A. Al-Kharsani,et al.  On the derivatives of a family of analytic functions , 2003 .

[18]  J. Dziok,et al.  On the convex combination of the Dziok-Srivastava operator , 2007, Appl. Math. Comput..

[19]  S. D. Bernardi,et al.  THE RADIUS OF UNIVALENCE OF CERTAIN ANALYTIC FUNCTIONS , 1966 .

[20]  Sukhjit Singh,et al.  Convolution properties of a class of starlike functions , 1989 .

[21]  Hari M. Srivastava,et al.  Certain Subclasses of Analytic Functions Associated with the Generalized Hypergeometric Function , 2003 .

[22]  R. J. Libera.  Some classes of regular univalent functions , 1965 .

[23]  Hari M. Srivastava,et al.  Certain properties of the Dziok-Srivastava operator , 2004, Appl. Math. Comput..

[24]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[25]  Sanford S. Miller,et al.  Differential Subordinations: Theory and Applications , 2000 .

[26]  H. Srivastava,et al.  Some subclasses of multivalent analytic functions involving the Dziok–Srivastava operator , 2008 .

[27]  Hari M. Srivastava,et al.  Classes of analytic functions associated with the generalized hypergeometric function , 1999, Appl. Math. Comput..

[28]  Hari M. Srivastava,et al.  A class of multivalently analytic functions associated with the Dziok–Srivastava operator , 2009 .

[29]  Hari M. Srivastava,et al.  A unified class of k-uniformly convex functions defined by the Dziok-Srivastava linear operator , 2007, Appl. Math. Comput..

[30]  Stephan Ruscheweyh.  NEW CRITERIA FOR UNIVALENT FUNCTIONS , 1975 .

[31]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .