Abstract
By introducing the function|lnx−lny|/(x+y+|x−y|), we establish new inequalities similar to Hilbert's type inequality for integrals. As applications, we give its equivalent form as well.
Highlights
If f, g are real functions such that 0 < ∞ 0 f 2(x)dx < ∞ and∞0 g2(x)dx < ∞, we have ∞ 0(x)g ( y ) x+y dxd y πf 2(x)dx g2(x)dx, (1.1)
Inequality (1.2) is named Hardy-Hilbert’s integral inequality, which is important in analysis and its applications, it has been studied and generalized in many directions by a number of mathematicians
If the constant 16 in (2.17) is not the best possible, by (2.20), we may get a contradiction that the constant factor in (2.1) is not the best possible
Summary
Where the constant factor π is the best possible. Inequality (1.1) had been generalized by Hardy-Riesz (see [2]) in 1925 as the following result. Where the constant factor c = π/ sin(π/ p) is the best possible. When p = q = 2, (1.2) reduces to (1.1). Inequality (1.2) is named Hardy-Hilbert’s integral inequality, which is important in analysis and its applications (see [3]), it has been studied and generalized in many directions by a number of mathematicians (see [4,5,6,7,8])
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