Sums and products from a finite set of real numbers

If A is a finite set of positive integers, letE(h)(A) denote the set of h-f old sums and h-fold products of elements of A. This paper is concerned with the behavior of the functionf(h)(k), the minimum of \E-h(A)\ taken over al l A with \A\ = k. Upper and lower bounds for f(h)(k) are proved, improving bounds given by Eros, Szemeredi, and Nathanson. Moreover, the lower bound h olds when we allow A to be a finite set of arbitrary positive real numbers.