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New Hilbert-Pachpatte type integral inequalities. (English) Zbl 0988.26013
This paper presents a new class of multivariable Hilbert type integral inequalities $$\int^{x_1}_0 \cdots \int^{x_n}_0 \frac{\prod^n_{i=0} |u^{(k_i)}_i (s_i)|ds_i \cdots ds_n}{\sum^n_{i=1} w_i s^{(\alpha_i +1)}_i /(q_i w_i)} \le U\prod^n _{i=1} x_i^{1/q_i}\prod^n_{i=1}\Biggl(\int^{x_i}_0 (x_i - s_i)^{(\beta_i +1)}\Phi_i (s_i)^{p_i} ds_i\Biggr)^{1/p_i},$$ where $$U = \frac 1{\prod^n_{i=1}[(\alpha_i + 1)^{1/q_i} (\beta_i + 1)^{1/p_i}]},$$ $u_i \in C^{m_i}([0,x_i])$, $\Phi_i \in C^{1}([0,x_i])$, $\Phi_i \ge 0$, $\alpha_i = (a_i +b_i q_i)(m_i - k_i - 1)$, $\beta_i = a_i (m_i -k_i - 1)$, $i \in I$ and $w_i \in \bbfR$, $w_i >0$, $\sum^n_{i=1} w_i = 1.$ Under suitable choices of the functions $\Phi_i$, the authors obtain this new class of related integral inequalities which generalizes recent results of {\it B. G. Pachpatte} [Tamkang J. Math. 30, No. 2, 139-146 (1999; Zbl 0962.26006); J. Math. Anal. Appl. 243, No. 2, 217-227 (2000; Zbl 0958.26013)].

26D10Inequalities involving derivatives, differential and integral operators
26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
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