Abstract：By introducing some parameters, we construct a non-homogeneous kernel function including hyperbolic functions on the whole plane. In addition, by using the rational fraction expansion of tangent function, a Hilbert-type integral inequality associated with the best possible constant factor and the higher derivatives of tangent function is presented. Furthermore, some meaningful and special results are presented by specializing the parameters with different values as on application.
有名辉, 孙霞. 一个R2上含双曲函数核的Hilbert型不等式[J]. 浙江大学学报（理学版）, 2020, 47(5): 554-558.
YOU Minghui, SUN Xia. A Hilbert-type inequality defined on R2 with the kernel involving hyperbolic functions. Journal of ZheJIang University(Science Edition), 2020, 47(5): 554-558.
1 HARDY G H, LITTLEWOOD J E, POLYA G． Inequalities[M]. London：Cambridge University Press, 1952． 2 杨必成．关于一个Hilbert类积分不等式的推广及应用[J]．应用数学，2003,16(2)：82-86． YANG B C．On a generalization of a Hilbert’s type integral inequality and its application[J]. Mathematics Applicata，2003,16(2): 82-86． 3 JIN J J. On Hilbert’s type inequalities[J]. Journal of Mathematical Analysis and Applications, 2008, 340(2)：932-942. DOI: 10.1016/j.jmaa.2007.09.036 4 MICHAEL T H, ASSIAS R, YANG B C. A Hilbert-type integral inequality in the whole plane related to the hypergeometric function and the beta function[J]. Journal of Mathematical Analysis and Applications, 428(1): 1286-1308. DOI: 10.1016/j.jmaa.2015.04.003 5 KUANG J C, DEBNATH L． On new generalizations of Hilbert’s inequality and their applications[J].Journal of Mathematical Analysis and Applications, 2000, 245(1)：248-265． DOI: 10.1006/jmaa.2000.6766 6 YOU M H．On a new discrete Hilbert-type inequality and its application[J]. Mathematical Inequalities & Applications, 2015,18 (4): 1575-1587. 7 有名辉. 一个与 Euler数有关的Hilbert型不等式的推广[J]. 浙江大学学报(理学版), 2016,43(2) :144-148. DOI:10.3785/j.issn.1008-9497.2016.02.004 YOU M H. Generalization of a Hilbert-type inequality related to Euler number[J]．Journal of Zhejiang University(Science Edition), 2016,43(2):144-148. DOI:10.3785/j.issn.1008-9497.2016.02.004 8 杨必成. 算子范数与Hilbert型不等式[M]. 北京: 科学出版社, 2009. YANG B C．The Norm of Operator and Hilbert-Type Inequalities[M]. Beijing: Science Press, 2009. 9 KRNI? M, PE?ARI? J, PERI? I, et al. Recent Advances in Hilbert-Type Inequalities[M]. Zagreb: Element Press, 2012. 10 MINTRINOVIC D S, PECARIC J E, FINK M． Inequalities Involving Functions and Their Integrals and Derivatives[M]. Boston: Kluwer Academic Press, 1991. 11 刘琼,龙顺潮. 一个核为双曲余割函数的Hilbert型积分不等式[J].数学学报(中文版),2013,56(1)：97-104. LIU Q, LONG S C. A new Hilbert-type integral inequality with the kernel of hyperbolic cosecant function[J]. Acta Mathematica Sinica (Chinese Series), 2013, 56(1): 97-104. 12 LIU Q. A Hilbert-type integral inequality with the mixed kernel and its applications [J]. Mathematica Applicata, 2015, 28(3):567-573. DOI: 10.4064/cm6572-1-2016 13 LIU Q, SUN W. A Hilbert-type integral inequality with the mixed kernel of multi-parameters [J]. Comptes Rendus Mathematique, 2013, 351(15): 605-611. DOI: 10.1016/j.crma.2013.09.001. 14 杨必成,陈强. 一个核为双曲正割函数的半离散Hilbert型不等式[J]. 西南师范大学学报(自然科学版), 2015,40(2) :26-32. DOI: 10.13718/j.cnki.xsxb.2015.02.006 YANG B C, CHEN Q．On a half-discrete Hilbert-type inequality with kernel related to hyperbolic secant function[J].Journal of Southwest China Normal University (Natural Science Edition), 2015,40(2) :26-32. DOI: 10.13718/j.cnki.xsxb.2015.02.006 15 菲赫金哥尔茨 ΓM. 微积分学教程(第2卷)[M]. 徐献瑜，冷生明，梁文琪译.北京：高等教育出版社，2006：695,397. FIKHTENGOLTS ΓM. Calculus Course (Volume Second)[M]．Translated by XU X Y，LENG S M，LIANG W Q. Beijing:Higher Education Press, 2006:695，397. 16 匡继昌. 常用不等式[M].第3版. 济南:山东科学技术出版社，2010:5. KUANG J C． Applied Inequalities [M]．3rd ed. Jinan:Shandong Science and Technology Press, 2010:5.
2020, Vol. 47 
