### On inequalities of Hilbert's type

Yongjin Li, Bing He
2007 Bulletin of the Australian Mathematical Society
By introducing the function l/(min{x,y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case. Jo then we have (see Hardy, Littlewood and Polya [4]) r Jo r ^Ĵ o Jo X + V Uo Jo where the constant factor •K is the best possible. Inequality (1.1) is
more » ... he well known Hilbert's inequality. Inequality (1.1) had been generalised by Hardy-Riesz (see [3]) in 1925 as: If f,g > 0,p > 1, (1/p) + (I/9) = 1, 0 < / f{x)dx < oo and 0 < f g«(x)dx Jo Jo < oo, then Jo Jo z + V sin(w/p)\J 0 r ni)dx , J o J o where the constant factor n/(sin(n/p)) is the best possible. When p = q = 2, (1.2) reduces to (1.1), Inequality (1.2) is Hardy-Hilbert's integral inequality, which is important in analysis and its applications (see [7] ). It has been studied and generalised in many directions by a number of mathematicians (see [1, 2, 6, 8, 10] ). Recently, by introducing some parameters, Yang (see [11] ) obtained the following inequalities: