The Kantorovich inequality under integral constraints (Q1323186)

Let \(f: [0,1]\to [m,M]\), where \(0< m< M<\infty\). If, in addition, \(\int^ 1_ 0 f^{-1}(x)dx\leq m^{-1}\), we write \(f\in A^ 1_{m,M}\). The authors proved that for any \(f\in A^ 1_{m,M}\), \[ \int^ 1_ 0 f^ 2(x)dx\leq (4m)^{-1}[M- N+ (M+ m)^ 2/M]\left(\int^ 1_ 0 f(x)dx\right)^ 2 \] and that the equality sign occurs iff the non-increasing rearrangement \(f^*\) of \(f\) is given by \(f^*(t)= M\) for \(t\in [0,m/M)\) and \(f^*(t)= mM/(m+ M)\) for \(t\in [m/M,1]\).