SCHWEITZER 1968 Problem 2

Let $a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that $$( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).$$

J. Suranyi