Some Wgh Inequalities for Univalent Harmonic Analytic Functions

ABSTRACT

In this paper, some Wgh inequalities for univalent harmonic analytic functions defined by Wright's generalized hypergeometric (Wgh) functions to be in certain classes are observed and proved. Some consequent results are also discussed.

In this paper, some Wgh inequalities for univalent harmonic analytic functions defined by Wright's generalized hypergeometric (Wgh) functions to be in certain classes are observed and proved. Some consequent results are also discussed.

KEYWORDS

Harmonic Functions, Harmonic Starlike Functions, Wright’s Generalized Hypergeometric Functions

Harmonic Functions, Harmonic Starlike Functions, Wright’s Generalized Hypergeometric Functions

Cite this paper

nullP. Sharma, "Some Wgh Inequalities for Univalent Harmonic Analytic Functions,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 464-469. doi: 10.4236/am.2010.16061.

nullP. Sharma, "Some Wgh Inequalities for Univalent Harmonic Analytic Functions,"

References

[1] J. Clunie and T. Sheil-Small, “Harmonic Univalent Functions,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, Vol. 9, 1984, pp. 3-26.

[2] J. M. Jahangiri, “Harmonic Functions Starlike in the Unit Disk,” Journal of Mathematical Analysis and Applications, Vol. 235, No. 2, 1999, pp. 470-477.

[3] O. P. Ahuja and J. M. Jahangiri, “Sakaguchi-Type Har- monic Univalent Functions,” Scientiae Mathematicae Japonicae, Vol. 59, 2004, pp. 239-244.

[4] H. ?. ?uney, “Sakaguchi-Type Harmonic Univalent Functions with Negative Coefficients,” International Journal of Contemporary Mathematical Sciences, Vol. 2, No. 10, 2007, pp. 459-463.

[5] M. Al. Shaqsi and M. Darus, “On Subclass of Harmonic Starlike Functions with Respect to K-Symmetric Points,” International Mathematical Forum, Vol. 2, No. 57, 2007, pp. 2799-2805.

[6] M. Al. Shaqsi and M. Darus, “On Harmonic Univalent Functions with Respect to K-Symmetric Points,” International Journal of Contemporary Mathematical Sciences, Vol. 3, No. 1-4, 2008, pp. 111-118.

[7] Z. G. Wang, C. Y. Gao and S. M. Yuan, “On Certain Subclasses of Close-to-Convex and Quasi-Convex Functions with Respect to K-Symmetric Points,” Journal of Mathematical Analysis and Applications, Vol. 322, No. 1, 2006, pp. 97-106.

[8] K. Sakaguchi, “On Certain Univalent Mapping,” Mathmatical Society of Japan, Vol. 11, 1959, pp. 72-75.

[9] O. P. Ahuja, “Planar Harmonic Convolution Operators Generated by Hypergeometric Functions,” Integral Transforms and Special Functions, Vol. 18, No. 3, 2007, pp. 165-177.

[10] O. P. Ahuja, “Harmonic Starlike and Convexity of Integral Operators Generated by Hypergeometric Series,” Integral Transforms and Special Functions, Vol. 20, No. 8, 2009, pp. 629-641.

[11] O. P. Ahuja and H. Silverman, “Inequalities Associating Hypergeomatric Functions with Planer Harmonic Mappings,” Journal of Inequalities in Pure and Applied Mathematics, Vol. 5, No. 4, 2004.

[12] M. K. Aouf and J. Dziok, “Distortion and Convolutional Theorems for Operators of Generalized Fractional Cal- culus Involving Wright Function,” Journal of Applied Analysis, Vol. 14, No. 2, 2008, pp. 183-192.

[13] M. K. Aouf and J. Dziok, “Certain Class of Analytic Functions Associated with the Wright Generalized Hypergeometric Function,” Journal of Mathematics and Applications, Vol. 30, 2008, pp. 23-32.

[14] J. Dziok and R. K. Raina, “Families of Analytic Functions Associated with the Wright Generalized Hypergeometric Function,” Demonstratio Mathematica, Vol. 37, No. 3, 2004, pp. 533-542.

[15] J. Dziok and R. K. Raina, “Some Results Based on First Order Differential Subordination with the Wright’s Generalized Hypergeometric Function,” Commentarii Mathematici Universitatis Sancti Pauli, Vol. 58, No. 2, 2009, pp. 87-94.

[16] J. Dziok, R. K. Raina and H. M. Srivastava, “Some Classes of Analytic Functions Associated with Operators on Hilbert Space Involving Wright’s Generalized Hypergeometric Function,” Proceedings of the Jangjeon Mathematical Society, Vol. 7, 2004, pp. 43-55.

[17] G. Murugusundaramoorthy and R. K. Raina, “On a Sub-class of Harmonic Functions Associated with the Wright’s Generalized Hypergeometric Functions,” Hacettepe Journal of Mathematics and Statistics, Vol. 38, No. 2, 2009, pp. 129-136.

[18] G. Murugusundaramoorthy and K. Vijaya, “A Subclass of Harmonic Functions Associated with Wright’s Hyper-geometric Functions,” Applied Mathematics, Vol. 1, No. 2, 2010, pp. 87-93.

[19] R. K. Raina, “Certain Subclasses of Analytic Functions with Fixed Argument of Coefficients Involving the Wright’s Function,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 22, No. 1, 2006, pp. 51-59.

[20] R. K. Raina, “On Generalized Wright’s Hypergeometric Functions and Fractional Calculus Operators,” East Asian Journal of Mathematics, Vol. 21, No. 2, 2005, pp. 191- 203.

[21] R. K. Raina and P. Sharma, “Harmonic Univalent Functions Associated with Wright’s Generalized Hyper- geometric Functions,” Integral Transform and Special Functions, Communicated for Publication.

[22] P. Sharma, “A Class of Multivalent Analytic Functions with Fixed Argument of Coefficients Involving Wright’s Generalized Hypergeometric Functions,” Bulletin of Mathematical Analysis and Applications, Vol. 2, No. 1, 2010, pp. 56-65.

[23] P. Sharma, “Multivalent Harmonic Functions Defined by M-Tuple Integral Operators,” Commentationes Mathematicae, Vol. 50, No. 1, 2010, pp. 87-101.

[24] E. M. Wright, “The Asymptotic Expansion of the Generalized Hypergeometric Function,” Proceedings London Mathematical Society, Vol. 46, No.1, 1946, pp. 389-408

[25] H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” Halsted Press, Ellis Horwood Limited, Hichester, 1984.

[26] A. A. Kilbas, M. Saigo and J. J. Trujillo, “On the Generalized Wright Function,” Fractional Calculus and Applied Analysis, Vol. 5, No. 4, 2002, pp. 437-460

[1] J. Clunie and T. Sheil-Small, “Harmonic Univalent Functions,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, Vol. 9, 1984, pp. 3-26.

[2] J. M. Jahangiri, “Harmonic Functions Starlike in the Unit Disk,” Journal of Mathematical Analysis and Applications, Vol. 235, No. 2, 1999, pp. 470-477.

[3] O. P. Ahuja and J. M. Jahangiri, “Sakaguchi-Type Har- monic Univalent Functions,” Scientiae Mathematicae Japonicae, Vol. 59, 2004, pp. 239-244.

[4] H. ?. ?uney, “Sakaguchi-Type Harmonic Univalent Functions with Negative Coefficients,” International Journal of Contemporary Mathematical Sciences, Vol. 2, No. 10, 2007, pp. 459-463.

[5] M. Al. Shaqsi and M. Darus, “On Subclass of Harmonic Starlike Functions with Respect to K-Symmetric Points,” International Mathematical Forum, Vol. 2, No. 57, 2007, pp. 2799-2805.

[6] M. Al. Shaqsi and M. Darus, “On Harmonic Univalent Functions with Respect to K-Symmetric Points,” International Journal of Contemporary Mathematical Sciences, Vol. 3, No. 1-4, 2008, pp. 111-118.

[7] Z. G. Wang, C. Y. Gao and S. M. Yuan, “On Certain Subclasses of Close-to-Convex and Quasi-Convex Functions with Respect to K-Symmetric Points,” Journal of Mathematical Analysis and Applications, Vol. 322, No. 1, 2006, pp. 97-106.

[8] K. Sakaguchi, “On Certain Univalent Mapping,” Mathmatical Society of Japan, Vol. 11, 1959, pp. 72-75.

[9] O. P. Ahuja, “Planar Harmonic Convolution Operators Generated by Hypergeometric Functions,” Integral Transforms and Special Functions, Vol. 18, No. 3, 2007, pp. 165-177.

[10] O. P. Ahuja, “Harmonic Starlike and Convexity of Integral Operators Generated by Hypergeometric Series,” Integral Transforms and Special Functions, Vol. 20, No. 8, 2009, pp. 629-641.

[11] O. P. Ahuja and H. Silverman, “Inequalities Associating Hypergeomatric Functions with Planer Harmonic Mappings,” Journal of Inequalities in Pure and Applied Mathematics, Vol. 5, No. 4, 2004.

[12] M. K. Aouf and J. Dziok, “Distortion and Convolutional Theorems for Operators of Generalized Fractional Cal- culus Involving Wright Function,” Journal of Applied Analysis, Vol. 14, No. 2, 2008, pp. 183-192.

[13] M. K. Aouf and J. Dziok, “Certain Class of Analytic Functions Associated with the Wright Generalized Hypergeometric Function,” Journal of Mathematics and Applications, Vol. 30, 2008, pp. 23-32.

[14] J. Dziok and R. K. Raina, “Families of Analytic Functions Associated with the Wright Generalized Hypergeometric Function,” Demonstratio Mathematica, Vol. 37, No. 3, 2004, pp. 533-542.

[15] J. Dziok and R. K. Raina, “Some Results Based on First Order Differential Subordination with the Wright’s Generalized Hypergeometric Function,” Commentarii Mathematici Universitatis Sancti Pauli, Vol. 58, No. 2, 2009, pp. 87-94.

[16] J. Dziok, R. K. Raina and H. M. Srivastava, “Some Classes of Analytic Functions Associated with Operators on Hilbert Space Involving Wright’s Generalized Hypergeometric Function,” Proceedings of the Jangjeon Mathematical Society, Vol. 7, 2004, pp. 43-55.

[17] G. Murugusundaramoorthy and R. K. Raina, “On a Sub-class of Harmonic Functions Associated with the Wright’s Generalized Hypergeometric Functions,” Hacettepe Journal of Mathematics and Statistics, Vol. 38, No. 2, 2009, pp. 129-136.

[18] G. Murugusundaramoorthy and K. Vijaya, “A Subclass of Harmonic Functions Associated with Wright’s Hyper-geometric Functions,” Applied Mathematics, Vol. 1, No. 2, 2010, pp. 87-93.

[19] R. K. Raina, “Certain Subclasses of Analytic Functions with Fixed Argument of Coefficients Involving the Wright’s Function,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 22, No. 1, 2006, pp. 51-59.

[20] R. K. Raina, “On Generalized Wright’s Hypergeometric Functions and Fractional Calculus Operators,” East Asian Journal of Mathematics, Vol. 21, No. 2, 2005, pp. 191- 203.

[21] R. K. Raina and P. Sharma, “Harmonic Univalent Functions Associated with Wright’s Generalized Hyper- geometric Functions,” Integral Transform and Special Functions, Communicated for Publication.

[22] P. Sharma, “A Class of Multivalent Analytic Functions with Fixed Argument of Coefficients Involving Wright’s Generalized Hypergeometric Functions,” Bulletin of Mathematical Analysis and Applications, Vol. 2, No. 1, 2010, pp. 56-65.

[23] P. Sharma, “Multivalent Harmonic Functions Defined by M-Tuple Integral Operators,” Commentationes Mathematicae, Vol. 50, No. 1, 2010, pp. 87-101.

[24] E. M. Wright, “The Asymptotic Expansion of the Generalized Hypergeometric Function,” Proceedings London Mathematical Society, Vol. 46, No.1, 1946, pp. 389-408

[25] H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” Halsted Press, Ellis Horwood Limited, Hichester, 1984.

[26] A. A. Kilbas, M. Saigo and J. J. Trujillo, “On the Generalized Wright Function,” Fractional Calculus and Applied Analysis, Vol. 5, No. 4, 2002, pp. 437-460