A reversed Poincaré inequality for monotone functions
Using a result of \textit{G. Talenti} [Ann. Mat. Pura Appl., IV. Ser. 110, 353-372 (1976; Zbl 0353.46018)] according to which the boundary value problem: \[ -{{d}\over{dx}}\left(\left |{{du}\over{dx}}\right |^{n-2}{{du}\over{dx}}\right) =\lambda |u|^{m-2}u, \;u'(0)=u(1)=0, \;u(0)=b>0, \;n>1,\;m\geq 1, \] has a unique positive solution if and only if \[ \lambda =\lambda (b):=B(1/m,1-1/n)^n(n-1)b^{n-m}/n\cdot m^{n-1} \] where \(B(n,m)=\Gamma (n)\Gamma (m)/\Gamma (n+m)\) is the \textit{Beta function}, the authors prove that a nondecreasing \(C^1\)-function \(g(.):(0,b)\to [0,\infty)\) satisfies the inequality: \[ \left(\int_0^b(g'(u))^{{{1}\over{q}}} du\right)^q\leq {{b^{q-m-1}}\over {m^{q-2}}}B\left ({{1}\over {m}},{{q-2}\over {q-1}}\right)^{q-1} \int_0^bg(u)u^{m-1} du \] for any \(m\geq 1\), \(q>2\).
scientific article; zbMATH DE number 1437138