Some new Poincaré-type inequalities (Q2721598)

In this paper interesting improvements and generalizations of the Poincaré-type integral inequality NEWLINE\[NEWLINE\lambda_0 \int_{\Omega}f^2 dx \leq \int_{\Omega}|\nabla f|^2 dx ,NEWLINE\]NEWLINE [see \textit{D. S. Mitrinovic}, ``Analytic inequalities'' (1970; Zbl 0199.38101)] involving many functions of several variables are established. In particular, the author gives an interesting proof of the following inequality NEWLINE\[NEWLINE\int_{\Omega}\prod_{\alpha}\mid f^{\alpha}\mid^{q_{\alpha}} \leq \frac 1{n}\sum_{\alpha}\frac{q_{\alpha}}{p_{\alpha}}\Big(\frac{M}{2}\Big)^{p_{\alpha}} \int_{\Omega}|\nabla f^{\alpha}|^{p_\alpha},NEWLINE\]NEWLINE where \(f^{\alpha} \in C^{1}_{0}(\Omega), \Omega = \prod^{n}_{i=1}[a_i,b_i]\) is a bounded region in \(\mathbb{R}^n, p_{\alpha} \geq 2, q_{\alpha} \geq 0, \sum_{\alpha}\frac{q_{\alpha}}{p_{\alpha}} = 1, \alpha = 1, \dots, m\) and \(M = \max\{ b_i - a_i, i = 1, \dots, n\}.\) Furthermore, several consequences of the results in form of corollaries and remarks are presented and their interconnection with related results in the literature are pointed out.