On the Cauchy-Schwarz inequality

We prove that the inequality [GRAPHICS] holds for all natural numbers n and for all real numbers x(k) and y(k) (k = 1,..., n) with 0 < x(1) less than or equal to x(2)/2 less than or equal to ... less than or equal to x(n)/n and 0 <y(n) less than or equal to y(n-1) less than or equal to ... less than or equal to y(1), if and only if alpha greater than or equal to 3/4 and beta greater than or equal to 1 - alpha. Inequality (*) with alpha = 3/4 and beta = 1/4 refines results given by Liu Zheng (J. Math. Anal. Appl. 218 (1998), 13-21) and the author (J. Math. Anal. Appl. 168 (1992), 596-604). (C) 1999 Academic Press.