A sharp multidimensional Bergh type inequality (Q2748473)

A function on \(\mathbb R^n_+\), \(n\in \mathbb N\), is said to be non-decreasing (non-increasing) if it is non-decreasing (non-increasing) in each variable. Let \(\alpha = (\alpha_1,\dots,\alpha_n)\) and \(\beta = (\beta_1, \dots, \beta_n)\) satisfy \(-\infty < \alpha_i < \beta_i < \infty\), \(i=1,\dots,n\). We write \(f\in Q^{\alpha}_{\beta}\) if \(f\) is a non-negative function on \(\mathbb R^n_+\) such that \(f(x_1,\dots,x_n)/(x_1^{\alpha_1} \dots x_n^{\alpha_n})\) is non-decreasing and \(f(x_1,\dots,x_n)/(x_1^{\beta_1}\dots x_n^{\beta_n})\) is non-increasing. NEWLINENEWLINENEWLINEThe main result of the paper reads as follows: If \(f\in Q^{\alpha}_{\beta}\), \(\gamma = (\gamma_1,\dots,\gamma_n)\), \(\alpha_i < \gamma_i < \beta_i\), \(i=1,\dots,n\), and \(0<p<q\leq \infty\), then NEWLINE\[NEWLINE \left(\int_{\mathbb R^n_+} \left( \frac{f(x_1,\dots,x_n)}{x_1^{\gamma_1} \dots x_n^{\gamma_n}}\right)^q \frac{dx_1\dots dx_n}{x_1\dots x_n}\right)^{1/q} \tag{\(*\)} NEWLINE\]NEWLINE NEWLINE\[NEWLINE \leq p^{\frac{n}{p}} q^{-\frac{n}{q}} \left( \prod^n_{i=1} \frac{(\gamma_i - \alpha_i)(\beta_i - \gamma_i)}{\beta_i - \alpha_i}\right)^{\frac 1{p} - \frac 1{q}} \left(\int_{\mathbb R^n_+} \left( \frac{f(x_1,\dots,x_n)}{x_1^{\gamma_1}\dots x_n^{\gamma_n}}\right)^p \frac{dx_1\dots dx_n}{x_1\dots x_n}\right)^{1/p}. NEWLINE\]NEWLINE Moreover, when NEWLINE\[NEWLINE f(x_1,\dots,x_n) = \prod^n_{i=1}\min \{x_i^{\alpha_i}, x_i^{\beta_i}\}, NEWLINE\]NEWLINE the equality holds in \((*)\). This result generalizes Bergh's inequality valid for quasi-concave functions on \((0,\infty)\) [cf. \textit{J. Bergh}, Math. Z. 215, No. 2, 205-208 (1994; Zbl 0804.26017)].