Proc. Gamb. Phil. Soc. (1968), 64, 1023 1023 POPS 64-120 Printed in Great Britain Generalization of Holder's and Minkowski's inequalities BY D. E. DAYKIN AND C. J. ELIEZER University of Malaya, Kuala Lumpur (Received 28 November 1966) 1. Introduction. In a recent paper (l) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfying for every pair of unequal values x x and x 2 ). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a l! ...,a n ;b 1 ,..., b n , the function L which had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (Sa6) a when x = 0 and (2a 2 ) (S6 2 ) when x = 1, and the property of convexity gives Cauchy's inequality. Again for sets (a), (b) and (c) of non-negative real numbers, the function (Za 1+ax b 1+ fi x c 1+ v x ) (Ld L+ y x b x+ax c 1+ ^ x ) (l l a 1+ fi x b 1+ y x c 1+ax ), where a, /?, y are real numbers such that a+fi+y = 0, is convex with minimum at x = 0, except if the sets (a), (6) and (c) are proportional. From the property of convexity, certain inequalities may be derived. In this paper we consider two convex functions which are related to Holder's and Minkowski's inequalities. We are enabled to give new elementary proofs of these inequalities, and to derive certain generalizations. 2. Suppose^J, q; a 1; ...,a n ; b lt ...,b n are positive numbers. We define the function H/P H /b q \ x \ l!a i)) • where each summation is from 1 to n, and d k = a k b k , k = 1,2,..., n. Then we have THEOREM 1. (i) / / (l/p) + (llq) < 1, then H(x) is a convex function (except if all the a's are equal, all the b's are equal, and a i 6 i is unity, in which case H(x) is a constant). (ii) // (lip) + (l/q) = 1, then H(x) is a convex function with minimum atx = 0 (except if the sets (aP) and (b Q ) are proportional in which case H(z) is constant for all x). 64-2