Inclusion properties of a subclass of analytic functions defined by an integral operator involving the Gauss hypergeometric function

Abstract In the present paper, we introduce and investigate a new subclass of analytic functions in the open unit disk U , which is defined by the convolution ( f μ ) −1  ∗  f ( z ), where f μ ( z ) ≔ ( 1 - μ ) z 2 F 1 ( a , b ; c ; z ) + μ z z 2 F 1 ( a , b ; c ; z ) ′ ( z ∈ U ; μ ≧ 0 ) . Several interesting properties including (for example) integral-preserving properties of this analytic function class are also considered.

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

[2]  Hari M. Srivastava,et al.  Subordinations for multivalent analytic functions associated with the Dziok–Srivastava operator , 2009 .

[4]  Jin-Lin Liu,et al.  The Noor Integral and Strongly Starlike Functions , 2001 .

[5]  Nak Eun Cho,et al.  MULTIPLIER TRANSFORMATIONS AND STRONGLY CLOSE-TO-CONVEX FUNCTIONS , 2003 .

[6]  Nak Eun Cho.  Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator , 2008 .

[7]  R. Parvatham,et al.  Some applications of differential subordination , 1985, Bulletin of the Australian Mathematical Society.

[8]  Nak Eun Cho,et al.  A Class of Nonlinear Integral Operators Preserving Double Subordinations , 2008 .

[9]  Nak Eun Cho.  Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator , 2008 .

[10]  W. Janowski,et al.  Some extremal problems for certain families of analytic functions I , 1973 .

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

[12]  Muhammad Aslam Noor,et al.  On Integral Operators , 1999 .

[13]  Roger W. Barnard,et al.  Applications of convolution operators to problems in univalent function theory. , 1980 .

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

[15]  Nak Eun Cho.  The Noor integral operator and strongly close-to-convex functions , 2003 .

[16]  Sanford S. Miller,et al.  On a Briot-Bouquet Differential Subordination , 1983 .

[17]  Hari M. Srivastava,et al.  Current topics in analytic function theory , 1992 .

[18]  J. A Kim,et al.  MAPPING PROPERTIES FOR CONVOLUTIONS INVOLVING HYPERGEOMETRIC FUNCTIONS , 2003 .

[19]  Hari M. Srivastava,et al.  Some inclusion properties of a certain family of integral operators , 2002 .

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

[21]  Stephan Ruscheweyh.  The Pólya-Schoenberg Conjecture , 1988 .

[22]  Hari M. Srivastava,et al.  Some applications of the generalized Libera integral operator , 1986 .

[23]  T. Sheil-Small,et al.  Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture , 1973 .

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

[25]  Sanford S. Miller,et al.  Differential subordinations and inequalities in the complex plane , 1987 .

[26]  Hitoshi Saitoh,et al.  On the Coefficients of Starlike Functions , 1987 .

[27]  A. Wiśniowska.  Geometric characterization of strongly starlike functions , 2000 .

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

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

[30]  Khalida Inayat Noor.  Integral operators defined by convolution with hypergeometric functions , 2006, Appl. Math. Comput..