A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane

ABSTRACT

In this paper, by introducing some parameters and estimating the weight coefficient, we give a new Hilbert’s inequality with the integral in whole plane and with a non-homogeneous and the equivalent form is given as well. The best constant factor is calculated by the way of Complex Analysis.

In this paper, by introducing some parameters and estimating the weight coefficient, we give a new Hilbert’s inequality with the integral in whole plane and with a non-homogeneous and the equivalent form is given as well. The best constant factor is calculated by the way of Complex Analysis.

Cite this paper

nullZ. Xie and Z. Zeng, "A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane,"*Advances in Pure Mathematics*, Vol. 1 No. 3, 2011, pp. 84-89. doi: 10.4236/apm.2011.13019.

nullZ. Xie and Z. Zeng, "A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane,"

References

[1] G. H. Hardy, J. E. Littlewood and G. Polya, “Inequalities,” Cambridge University Press, Cambridge, 1952.

[2] G. H. Hardy, “Note on a Theorem of Hilbert Concerning Series of Positive Terems,” Proceedings of London Mathematical Soci-ety, Vol. 23, No. 2, 1925, pp. 45-46.

[3] B. C. Yang, “A New Hilbert-Type Integral Inequality with a Parameters,” Journal of Henan University (Natural Science), Vol. 35, No. 4, 2005, pp. 4-8.

[4] Z. T. Xie and Z. Zeng, “A Hilbert-Type Integral Inequality whose Kernel is a Homogeneous Form of De-gree-3,” Journal of Mathematical Analysis Application, Vol 339, No. 1, 2008, pp. 324-331 doi:10.1016/j.jmaa.2007.06.059

[5] B. C. Yang, “A New Hilbert-Type Integral Inequality with Some Parameters,” Journal of Jilin University (Science Edition), Vol. 46, No. 6, 2008, pp. 1085-1090.

[6] B. C. Yang, “A Hilbert-Type Inter-gral Inequality with the Homogeneous Kernel of Real Num-ber-Degree,” Journal of Jilin University (Science Edition), Vol. 47, No. 5, 2009, pp. 887-892.

[7] Z. T. Xie and X. D. Liu, “A New Hilbert-Type Integral Inequality and its Reverse,” Journal of Henan University, (Science Edition), Vol. 39, No. 1, 2009, pp. 10-13.

[8] Z. Zeng and Z. T. Xie, “On a New Hil-bert-Type Integral Inequality with the Integral in Whole Plane,” Journal of Inequalities and Applications, Article ID 256796, 2010.

[9] Z. T. Xie, B. C. Yang and Z. Zeng, “A New Hil-bert- type Integral Inequality with the Homogeneous Kernel of Real Number-Degree,” Journal of Jilin University (Science Edition), Vol. 48, No. 6, 2010, pp. 941-945.

[10] Z. T. Xie and Z. Zeng, “On Generality of Hilbert’s Inequality with Best Con-stant Factor,” Natural Science Journal of Xiangtan University, Vol. 32, No. 3, 2010, pp. 1-4.

[11] Z. T. Xie and B. L. Fu, “A New Hilbert-Type Integral Inequality with a Best Constant Factor,” Journal of Wuhan University (Natural Science Edi-tion), Vol. 55, No. 6, 2009, pp. 637-640.

[12] Z. T. Xie and Z. Zeng, “The Hilbert-Type Integral Inequality with the System Kernel of Degree Homogeneous Form,” Kyungpook Mathematical Journal, Vol. 50, 2010, pp. 297-306.

[13] Z. T. Xie and F. M. Zhou, “A Generalization of a Hilbert-Type Ine-quality with the Best Constant Factor,” Journal of Sichuan Normal University (Natural Science), Vol. 32, No. 5, 2009, pp. 626-629.

[14] Z. T. Xie and Z. Zeng, “A Hilbert-Type Integral Inequality with a Non-Homogeneous Form and a Best Constant Factor,” Advances and Applications in Mathematical Sciens, Vol. 3, No. 1, 2010, pp. 61-71.

[15] Z. Zeng and Z. T. Xie, “A New Hilbert-Type Integral Inequality with a Best Constant Factor,” Journal of South China Normal University (Natural Science Edition), Vol. 3, 2010, pp. 31-33.

[1] G. H. Hardy, J. E. Littlewood and G. Polya, “Inequalities,” Cambridge University Press, Cambridge, 1952.

[2] G. H. Hardy, “Note on a Theorem of Hilbert Concerning Series of Positive Terems,” Proceedings of London Mathematical Soci-ety, Vol. 23, No. 2, 1925, pp. 45-46.

[3] B. C. Yang, “A New Hilbert-Type Integral Inequality with a Parameters,” Journal of Henan University (Natural Science), Vol. 35, No. 4, 2005, pp. 4-8.

[4] Z. T. Xie and Z. Zeng, “A Hilbert-Type Integral Inequality whose Kernel is a Homogeneous Form of De-gree-3,” Journal of Mathematical Analysis Application, Vol 339, No. 1, 2008, pp. 324-331 doi:10.1016/j.jmaa.2007.06.059

[5] B. C. Yang, “A New Hilbert-Type Integral Inequality with Some Parameters,” Journal of Jilin University (Science Edition), Vol. 46, No. 6, 2008, pp. 1085-1090.

[6] B. C. Yang, “A Hilbert-Type Inter-gral Inequality with the Homogeneous Kernel of Real Num-ber-Degree,” Journal of Jilin University (Science Edition), Vol. 47, No. 5, 2009, pp. 887-892.

[7] Z. T. Xie and X. D. Liu, “A New Hilbert-Type Integral Inequality and its Reverse,” Journal of Henan University, (Science Edition), Vol. 39, No. 1, 2009, pp. 10-13.

[8] Z. Zeng and Z. T. Xie, “On a New Hil-bert-Type Integral Inequality with the Integral in Whole Plane,” Journal of Inequalities and Applications, Article ID 256796, 2010.

[9] Z. T. Xie, B. C. Yang and Z. Zeng, “A New Hil-bert- type Integral Inequality with the Homogeneous Kernel of Real Number-Degree,” Journal of Jilin University (Science Edition), Vol. 48, No. 6, 2010, pp. 941-945.

[10] Z. T. Xie and Z. Zeng, “On Generality of Hilbert’s Inequality with Best Con-stant Factor,” Natural Science Journal of Xiangtan University, Vol. 32, No. 3, 2010, pp. 1-4.

[11] Z. T. Xie and B. L. Fu, “A New Hilbert-Type Integral Inequality with a Best Constant Factor,” Journal of Wuhan University (Natural Science Edi-tion), Vol. 55, No. 6, 2009, pp. 637-640.

[12] Z. T. Xie and Z. Zeng, “The Hilbert-Type Integral Inequality with the System Kernel of Degree Homogeneous Form,” Kyungpook Mathematical Journal, Vol. 50, 2010, pp. 297-306.

[13] Z. T. Xie and F. M. Zhou, “A Generalization of a Hilbert-Type Ine-quality with the Best Constant Factor,” Journal of Sichuan Normal University (Natural Science), Vol. 32, No. 5, 2009, pp. 626-629.

[14] Z. T. Xie and Z. Zeng, “A Hilbert-Type Integral Inequality with a Non-Homogeneous Form and a Best Constant Factor,” Advances and Applications in Mathematical Sciens, Vol. 3, No. 1, 2010, pp. 61-71.

[15] Z. Zeng and Z. T. Xie, “A New Hilbert-Type Integral Inequality with a Best Constant Factor,” Journal of South China Normal University (Natural Science Edition), Vol. 3, 2010, pp. 31-33.