ON AN INTEGRAL INEQUALITY OF R. BELLMAN
ON AN INTEGRAL INEQUALITY OF R. BELLMAN

 
 
 We prove: if $u$ and $v$ are non-negative, concave functions defined on $[0, 1]$ satisfying 
 \[\int_0^1 (u(x))^{2p} dx =\int_0^1 (v(x))^{2q} dx=1, \quad p>0, \quad q>0,\]
 then
 \[\int_0^1(u(x))^p (v(x))^q dx\ge\frac{2\sqrt{(2p+1)(2q+1)}}{(p+1)(q+1)}-1.\]