MONOTONICITY OF SEQUENCES INVOLVING CONVEX AND CONCAVE FUNCTIONS

摘要

Let f be an increasing and convex (concave) function on [0, 1) and φ a positive increasing concave function on [0,∞) such that φ(0)=0 and the sequence {φ(i+1) (φ(i+1)/φ(i) - 1)}_(i∈N) decreases (the sequence {φ(i)(φ(i)/φ(i+1) - 1)}_(i∈N increases).Then the sequence {1/φ(n) ∑_(i=0)~(n-1) f(φ(i)/φ(n))}_(n∈N) is increasing.