Some Monotonicity Properties of Convex Functions with Applications

Abstract

We mainly establish a monotonicity property between some special Riemann sums of a convex function $f$ on $[a,b]$, which in particular yields that $\frac{b-a}{n+1}{\sum }_{i=0}^{n}f\left(a+i\frac{b-a}{n}\right)$ is decreasing while $\frac{b-a}{n-1}{\sum }_{i=1}^{n-1}f\left(a+i\frac{b-a}{n}\right)$ is an increasing sequence. These give us a new refinement of the Hermitt-Hadamard inequality. Moreover, we give a refinement of the classical Alzer's inequality together with a suitable converse to it. Applications regarding to some important convex functions are also included.

Cited by in corpus (1)

• On the monotonicity of left and right Riemann sums