Prolegomena To a Chapter on Inequalities

I take this opportunity of correcting three small errors or omissions in my address with this title * . (1) On p. 68 I say “I can find no such proof [of the inequality x p −1 > p ( x −1)] in the books”. Dr. Poole refers me to Stolz und Gmeiner, Theoretische Arithmetic , II (1902), 202–208, where there is a proof, less concise than that which I give, but conforming to the criteria laid down in the address. (2) The proofs of Hölder's inequality, for sums and integrals, given on pp. 67–68 and 71, were based on proofs contained in the printed but unpublished notes of Prof. Littlewood's lectures, which havo been in my possession for some years. They are, however, the same in principles as the proof of the inequality for integrals given by F. Riesz, “Su alcune disuguaglianze”, Bollettino dell' Unione Mat. Italiana , 7 (1928), 77–79. (3) Prof. Pólya points out to me that one of my remarks on p. 70 is hardly justifiable, since the inequality for integrals of bounded functions may be deduced from that for sums as follows. If we suppose, for simplicity, that f and g lie always in (0, 1), and denote by e rs the measure of the set in whichr - 1 n > f ⩽ r n , s - 1 n > f ⩽ s nWe have∫ 0 1 f g   d x = lim 1 n 2∑ ∑ r s   e r s ⩽( lim 1 n 2∑ ∑r 2   e r s)1 2( lim 1 n 2∑ ∑s 2e r s)1 2=( ∫ 0 1f 2 d x )1 2( ∫ 0 1g 2 d x )1 2The argument uses double sums, but there is no relevant difference between a finite double sum and an ordinary sum.