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On some new inequalities similar to Hilbert’s inequality. (English) Zbl 0911.26012
Let $p,q\in [1,\infty)$, $k,r\in\bbfN$, $\{a_m\}^k_1,\{b_n\}^r_1\subset [0,\infty)$, $A_m= \sum^m_{s=1} a_s$ and $B_n= \sum^n_{t=1} b_t$. Then $$\sum^k_{m= 1} \sum^r_{n= 1} {A^p_m B^q_n\over m+n}\le C\Biggl(\sum^k_{m= 1} (k- m+1) (A^{p-1}_m a_m)^2\Biggr)^{1/2} \Biggl(\sum^r_{n= 1}(r- n+1) (B^{q- 1}_n b_n)^2\Biggr)^{1/2}$$ (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.
Reviewer: B.Opic (Praha)

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
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