On some new inequalities similar to Hilbert's inequality
Let \(p,q\in [1,\infty)\), \(k,r\in\mathbb{N}\), \(\{a_m\}^k_1,\{b_n\}^r_1\subset [0,\infty)\), \(A_m= \sum^m_{s=1} a_s\) and \(B_n= \sum^n_{t=1} b_t\). Then \[ \sum^k_{m= 1} \sum^r_{n= 1} {A^p_m B^q_n\over m+n}\leq C\Biggl(\sum^k_{m= 1} (k- m+1) (A^{p-1}_m a_m)^2\Biggr)^{1/2} \Biggl(\sum^r_{n= 1}(r- n+1) (B^{q- 1}_n b_n)^2\Biggr)^{1/2} \] (unless \(\{a_m\}^k_1\subset \{0\}\) or \(\{b_n\}^r_1\subset \{0\}\)), where \(C= 2^{-1} pq\sqrt{kr}\). This is one of the main results of the paper, the others are its modifications. Note that integral analogues of the mentioned results are also proved.
scientific article; zbMATH DE number 1216737