Sequence of real numbers $a_0,a_1,\dots,a_{n}$ are called concave if for each $0<i<n$, $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.$a_{0}=0$ Find the largest $c(n)$ such that for every concave sequence of non-negative real numbers:
such
$$c(n)\sum_{i=1}^{n}a^2_{i}\le \left(\sum_{i=1}^{n}a_{i}\right)^2$$
I know $c(n)=\dfrac{n-1}{2}$ is well Khinchine inequality.but for best I can't it