Let a(o) < a(1) <...< a(n) be positive integers with sums Sigma(n)(i=o) e(i)a(i)(epsilon(i) = 0, 1) distinct. P. Erdos conjectured that E(i=0)(n) 1/a(i) <= E(i=o)(n) 1/2(i). The best known result along this line is that Chen: Let f be any given convex decreasing function on [A, B] with alpha(0), alpha(1),...,alpha(n), beta(0), beta(1),...,beta(n) being real numbers in [A, B] with alpha(0) <= alpha(1) <=... <= alpha(n), Sigma(k)(i=0) alpha(i) >= Sigma(k)(i=0) beta(i), k = 0,...,n. Then Sigma(n)(i=o), f(alpha(i)) <= Sigma(n)(i=o) f(beta(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 alpha(0), alpha(1),...,alpha(n), beta(0), beta(1),..., beta(n), alpha(0)', alpha(1)',..., alpha(n)', beta(0)', beta(1)',...,beta(n)' be real numbers in [A, B] with alpha(0) <= alpha(1) <= ... <= alpha(n), Sigma(t)(i=o) alpha(i) >= Sigma(t)(i=o), beta(i), t = 0,..., n,alpha'(o), <= alpha'(1) <=... <= alpha'(n), Sigma(t)(i=o)< alpha'(i) >= Sigma(t)(i=0) beta'(i),t = 0,...,n. Then Sigma(n)(i=0) f(alpha(i))g(alpha'(i)) <= Sigma(n)(i=0)f(beta(i))g(beta'(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 alpha(0), alpha(1),..., alpha(n), beta(0), beta(1),..., beta(n) being real numbers in [A, B] with alpha(0) <= alpha(1) <= ...<= alpha(n), Sigma(t)(i=0) k(i)alpha(i) >= Sigma(t)(i=0) k(i)beta(i), t = 0,..., n. Then Sigma(n)(i=0) k(i) f (alpha(i)), <= Sigma(n)(n=0) k(i)f(beta(i)).
