ILL 1974 Problem 32

Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$ . If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$ , prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.