A new generalization of Hardy-Hilbert's inequality with non-homogeneous kernel

Abstract

Let p > 1 , 1/p + 1/p∗ = 1 , and a = (an)n=1 , b = (bm) ∞ m=1 be two complex sequences. We exhibit the generalization of Hardy-Hilbert’s inequality of the following type: ∑ n,m 1 K(φ1(n),φ2(m))|an||bm| < C ( ∞ ∑ n=1 | an f1(φ1(n)) |p ) 1 p ( ∞ ∑ m=1 | bm f2(φ2(m)) |p ) 1 p∗ , where K : (0,∞)× (0,∞) → (0,∞) , f1, f2,φ1,φ2 : (0,∞) → (0,∞) and C is a suitable constant. We also get several interesting inequalities which generalize many recent results. Mathematics subject classification (2010): 26D15, 47B37.