Two Inequalities for Convex Functions

Let a 0 > a 1 > > a n be positive integers with sumsni=0iai(i=0,1)distinct. P. Erds conjectured that ni=01/aini=01/2i.The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with 0, 1, ... , n , 0, 1, ... , n being real numbers in [A, B] with 0 1 n , ki=0iki=0i,k=0,,n.Then ni=0f(i)ni=0f(i). In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and 0, 1, ... , n , 0, 1, ... , n , ' 0, ' 1, ... , ' n , ' 0 , ' 1 , ... , ' n be real numbers in [A, B] with 0 1 n , ti=0iti=0i,t=0,,n, ' 0 ' 1 ' n , ti=0iti=0i,t=0,,n.Then ni=0f(i)g(i)ni=0f(i)g(i). Theorem 2 Let f be any given convex decreasing function on [A, B] with k 0, k 1, ... , k n being nonnegative real numbers and 0, 1, ... , n , 0, 1, ... , n being real numbers in [A, B] with 0 1 n , ti=0kiiti=0kii,t=0,,n. Then ni=0kif(i)ni=0kif(i).