TAILIEUCHUNG - Note on Hilbert-type inequalities
The main objective of this paper is to prove Hilbert-type and Hardy-Hilbert-type inequalities with a general homogeneous kernel, thus generalizing a result obtained in Namita Das and Srinibas Sahoo, A generalization of multiple Hardy-Hilbert’s integral inequality, Journal of Mathematical Inequalities. | Turk J Math 36 (2012) , 253 – 262. ¨ ITAK ˙ c TUB doi: Note on Hilbert-type inequalities Predrag Vukovi´c Abstract The main objective of this paper is to prove Hilbert-type and Hardy-Hilbert-type inequalities with a general homogeneous kernel, thus generalizing a result obtained in [Namita Das and Srinibas Sahoo, A generalization of multiple Hardy-Hilbert’s integral inequality, Journal of Mathematical Inequalities, 3(1), (2009), 139–154]. 1. Introduction Bicheng Yang in [4] proved a Hilbert-type inequality for conjugate parameters and with the kernel K(x, y) = (u(x) + u(y))−s , s > 0. His result is contained in the following theorem. 1 p Theorem A If p > 1, + 1 q = 1, φr > 0 (r = p, q ), φp + φq = s, u(t) is a differentiable strict increasing function in (a, b) (−∞ ≤ a 0 (r = p, q), φp +φq = s} = ∅, with the above assumption, one has the reverse of (), and the constant is still the best possible. Recently, Namita Das et al. [1] gave a generalization of Yang’s result: n 1 Theorem B Let n ∈ N\{1}, pi > 1, (i = 1, 2, . . . , n), i=1 pi = 1, s > 0, λi > 0 (i = 1, 2, . . . , n) with n i=1 λi = s. Suppose for every = 1, . . . , n; ui : (ai , bi ) → (0, ∞), is a strictly increasing differentiable function such that ui (ai ) = 0 and ui (bi ) = ∞. If fi ≥ 0 (j = 1, 2, . . . , n), satisfy bj 0 0. To obtain the main results we define the function k (β1 , . . . , βn−1 ) by β k (β1 , . . . , βn−1 ) := (0,∞)n−1 n−1 K(1, t1 . . . , tn−1)tβ1 1 · · · tn−1 dt1 · · · dtn−1 , () where we suppose that k (β1 , . . . , βn−1 ) −1 and β1 + · · · + βn−1 + n 1 , i = 1, . . . , n, and let Let K : (0, ∞) n 1 q = n−1 1 i=1 pi . → R be non-negative measurable homogeneous function of degree −s, s > 0 , and let Aij , i, j = 1, . . . , n, and αi , i = 1, . . . , n be real parameters satisfying () and (). If fi : (0, ∞) → R , fi = 0 , i = 1, . . . , n are non-negative measurable functions, then the following inequalities hold and .
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