Question 5
Let $t_1,\ldots,t_n$ be real numbers with $t_k\in[p,q]$ for $k=1,\ldots,n$, where $0\lt p\le q$ are given. Define
$$A=\frac1n(t_1+\cdots+t_n),\qquad B=\frac1n(t_1^2+\cdots+t_n^2).$$
Prove that $\dfrac{A^2}{B}\ge\dfrac{4pq}{(p+q)^2}$ and determine the necessary and sufficient condition for equality.
$$A=\frac1n(t_1+\cdots+t_n),\qquad B=\frac1n(t_1^2+\cdots+t_n^2).$$
Prove that $\dfrac{A^2}{B}\ge\dfrac{4pq}{(p+q)^2}$ and determine the necessary and sufficient condition for equality.