On the Cauchy-Schwarz Inequality

We prove that the inequality n 2 n n b xkyk F yk a q xk2yk ). / /k ks1 ks1 ks1 holds for all natural numbers n and for all real numbers xkand yk ks1, . . . , n. with 0-x1Fx2r2F ??? Fxnrn and 0-ynFyny1F ??? Fy1, if and only if a G3r4 and b G1ya . Inequality ). with a s3r4 and b s1r4 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..