On a Hilbert-type integral inequality in the whole plane

Document Type : Research Article

Authors

1 Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece

2 Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P. R. China

3 University of Ioannina‎, ‎Ioannina‎, ‎Greece.

Abstract

Using weight functions and techniques of real analysis, a new Hilbert-type integral inequality in the whole plane with nonhomogeneous kernel and a best possible constant factor is proved. Equivalent forms, several particular inequalities and operator expressions are considered.

Keywords

Main Subjects


  1. [1] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, USA, 1934.

    [2] B.C. Yang, The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijing, China, 2009.

    [3] B.C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers Ltd., The United Arab Emirates, 2009

    [4] B.C. Yang, On the norm of an integral operator and applications, J. Math. Anal. Appl. 321 (2006), no. 1, 182–192.

    [5] B.C. Yang, On the norm of a Hilbert’s type linear operator and applications, J. Math. Anal. Appl. 325 (2007), no.

    1, 529–541.

    [6] B.C. Yang, A Hilbert-type integral inequality with a homogeneous kernel of degree zero, (Chinese) J. Shandong

    Univ. Nat. Sci. 45 (2010), no. 2, 103–106.

    [7] L. Debnath, B.C. Yang, Recent developments of Hilbert-type discrete and integral inequalities with applications,

    Int. J. Math. Math. Sci. 2012, Art. ID 871845, 29 pages.

    [8] M. Th. Rassias, B.C. Yang, On half-discrete Hilbert’s inequality, Appl. Math. Comput. 220 (2013) 75–93.

    [9] B.C. Yang, M. Krnić, A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0, J.

    Math. Inequal. 6(2012), no. 3, 401–417.

    [10] M. Krnić, P. Vuković, Multidimensional Hilbert-type inequalities obtained via local fractional calculus, Acta Appl.

    Math. 169 (2020), 667–680. doi:10.1007/s10440-020-00317-x

    [11] V. Adiyasuren, T. Batbold, and M. Krnić, Half-discrete Hilbert-type inequalities with mean operators, the best

    constants, and applications, Applied Mathematics and Computation, 231 (2014), 148–159 doi:10.1016/j.amc.

    2014.01.011

    [12] V. Adiyasuren, T. Batbold, and M. Krnić, On several new Hilbert-type inequalities involving means operators, Acta

    Math. Sin. (Engl. Ser.) 29 (2013), no. 8, 1493–1514.

    [13] I. Brnetić, M. Krnić, and J. Peˇcarić, Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters,

    Bull. Austral. Math. Soc. 71 (2005), no. 3, 447–457.

    [14] Th.M. Rassias, B.C. Yang, A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function,

    Appl. Math. Comput. 225 (2013), 263–277.

    [15] M.Th. Rassias, B.C. Yang, On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic

    cotangent function, Appl. Math. Comput. 242 (2014), 800–813.

    [16] C.J. Zhao, W.S. Cheung, On Hilbert’s inequalities with alternating signs, J. Math. Inequal 12 (2018), no. 1, 191–200.

    [17] P. Gao, On weighted Hardy inequalities for non-increasing sequences, J. Math. Inequal. 12 (2018), no. 2, 551–557.

    [18] M. Th. Rassias and B.C. Yang, On an equivalent property of a reverse Hilbert-type integral inequality related to

    the extended Hurwitz-zeta function, J. Math. Inequal. 13 (2019), no. 2, 315–334.

    [19] B.C. Yang, A new Hilbert-type integral inequality, Soochow J. Math. 33 (2007), no. 4, 849–859.

    [20] Z.Q. Wang, D.R. Guo, Introduction to special functions, Science Press, Beijing, China, 1979.

    [21] B. He, B.C. Yang, On a Hilbert-type integral inequality with the homogeneous kernel of 0-degree and the hypergeometrc

    function, Mathematics in Practice and Theory 40 (2010), no. 18, 105–211.

    [22] B.C. Yang, A new Hilbert-type integral inequality with some parameters, J. Jilin Univ. (Science Edition) 46 (2008),

    1. 6, 1085–1090.

    [23] B.C. Yang, A Hilbert-type integral inequality with a nonhomogeneous kernel, (Chinese) Xiamen Daxue Xuebao

    Ziran Kexue Ban 48 (2009), no. 2, 165–169.

    [24] B.C. Yang, A reverse Hilbert-type integral inequality with some parameters, J. Xinxiang Univ. (Natural Science

    Edition), 27 (2010), no. 6, 1–4.

    [25] D.M. Xin, B.C. Yang, A Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree

    2, J. Inequal. Appl. 2011, Art. ID 401428, 11 pages.

    [26] B. Yang, A reverse Hilbert-type integral inequality with a nonhomogeneous kernel, (Chinese) J. Jilin Univ. Sci. 49

    (2011), no. 3, 437–441.

    [27] Z.T. Xie, Z. Zeng, Y.F. Sun, A new Hilbert-type inequality with the homogeneous kernel of degree-2, Adv. Appl.

    Math. Sci. 12 (2013), no. 7, 391–401.

    [28] Z. Zhen, K. Raja Rama Gandhi, Z.T. Xie, A new Hilbert-type inequality with the homogeneous kernel of degree -2

    and with the integral, Bull. Math. Sci. App. 3 (2014), no. 1, 11–20.

    [29] M.Th. Rassias, B.C. Yang,A Hilbert-type integral inequality in the whole plane related to the hypergeometric

    function and the beta function, J. Math. Anal. Appl. 428 (2015), no. 2, 1286–1308.

    [30] Z.H. Gu, B.C. Yang, A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a

    few parameters, J. Inequal. Appl. (2015), 2015:314, 9 pages.

    [31] M.Th. Rassias, B.C. Yang, On a Hilbert-type integral inequality in the whole plane related to the extended

    Riemann zeta function, Complex Anal. Oper. Theory 13 (2019), no. 4, 1765–1782. https://doi.org/10.1007/

    s11785-018-0830-5.

    [32] M.Th. Rassias, B.C. Yang, A. Raigorodskii,On Hardy-type integral inequalities in the whole plane related to the

    extended Hurwitz-zeta function, J. Inequal. Appl. (2020), Paper No. 94, 24 pages.

    [33] M.Th. Rassias, B.C. Yang, A. Raigorodskii, On the reverse Hardy-type integral inequalities in the whole plane with

    the extended Riemann-zeta function, J. Math. Inequal. 14 (2020), no. 2, 525–546.

    [34] M.Th. Rassias, B.C. Yang, A. Raigorodskii, On a more accurate reverse Hilbert-type inequality in the whole plane,

    1. Math. Inequal. 14 (2020), no. 4, 1359–1374.

    [35] M.Th. Rassias, B.C. Yang, A. Raigorodskii, A Hilbert-type integral inequality in the whole plane related to the Arc

    tangent function, Symmetry 2021, 13, 351. https://doi.org/10.3390/sym13020351.

    [36] M.Th. Rassias, B.C. Yang, G.C.Meletiou, A more accurate half-discrete Hilbert-type inequality in the whole plane

    and the reverses, Ann. Funct. Anal. (2021), no. 3, Paper No. 50, 29 pages.

    [37] J.C. Kuang, Applied inequalities, Shangdong Science and Technology Press, Jinan, China, 2004.

    [38] J.C. Kuang, Introudction to Real Analysis, Hunan Educiton Press, Changsha, China, 1996.