On further strengthened Hardy-Hilbert's inequality

Abstract

We obtain an inequality for the weight coefficient ω ( q , n ) ( 1$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"> q > 1 , 1 / q + 1 / q = 1 , n ∈ ℕ ) in the form ω ( q , n ) = : ∑ m = 1 ∞ ( 1 / ( m + n ) ) ( n / m ) 1 / q π / sin ( π / p ) − 1 / ( 2 n 1 / p + ( 2 / a ) n − 1 / q ) where 0 a 147 / 45 , as n ≥ 3 ; 0 a ( 1 − C ) / ( 2 C − 1 ) , as n = 1 , 2 , and C is an Euler constant. We show a generalization and improvement of Hilbert's inequalities. The results of the paper by Yang and Debnath are improved.