On a polynomial inequality of P. Erdős and T. Grünwald (Q1300282)

Let \(C_n:= (-1,1)\times\cdots\times (-1,1)\) be an open \(n\)-dimensional cube and let \(\Gamma_n\) be the class of functions of \(n\)-variables \(x_1,\dots, x_n\) which are of the form \[ f(_1,\dots, x_n):= (1- x^2_1)\cdots(1- x^2_n) \psi(x_1,\dots, x_n), \] where \(|\psi|\) is logarithmically concave on \(C_n\). The main result of this paper is theorem 2. Let a function \(f\in\Gamma_n\) be such that \(|f(x_1,\dots, x_n)|\leq 1\) in \(C_n\). Then for all \(p>0\), \[ \begin{multlined} \Biggl(2_n \int\cdots\int_{C_n} |f(x_1,\dots, x_n)|^p dx_1\cdots dx_n\Biggr)^{1/p}<\\ \Biggl(2_1 \int^1_{-1}|1-x^2|^p dx\Biggr)^{n/p} \sup_{(x_1,\dots, x_n)\in C_n}|f(x_1,\dots, x_n)|,\end{multlined} \] unless the function \(f(x_1,\dots, x_n)\neq (1- x^2_1)\cdots(1- z^2_n)\). This result is the extension of the inequality of P. Erdős and T. Grünwald for polynomials with only zeros having \(-1\), \(1\) as consecutive zeros even for the case \(n=1\).