MathDB

Let

f:[0,\infty)\to\mathbb{R}

be a differentiable function, with a continous derivative. Given that

f(0)=0

and

0\leqslant f'(x)\leqslant 1

for every

x>0

prove that

\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,

for any positive integer

n{}

and real number

a>0.

Problem Statement