Carleman's Inequality

Abstract

The statement of the inequality is actually a little more precise, but we shall discuss that later. Carleman used the inequality as part of a new proof of a delightful theorem of Denjoy about quasi-analytic functions. We shall not pursue that application here since we are interested in the inequality as a theorem about infinite series. We wish to expound in detail some of the many proofs of this theorem. We remark in passing, with approval, that editors of mathematical journals in the 1920s were happy to publish many different proofs of the same theorem. There are many known results about the convergence (or divergence) of series obtained by linear transformations of a nonnegative sequence {an }. For example, if Ain is the arithmetic mean of a1, a2, ... , an, then Y An, always diverges unless {an } is the zero sequence (for the sum dominates part of the harmonic series). The arithmeticgeometric mean inequality (AGM) asserts that y, < An, with equality if and only if the sequence {an} is constant. So the AGM-inequality must be strict when E an is convergent and nonzero. It is perhaps surprising a priori that the AGM-inequality is so