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arXiv · 1310.6698

Some Monotonicity Properties of Convex Functions with Applications

Jamal Rooin·Hossein Dehghan
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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.

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Jamal Rooin, Hossein Dehghan. 2013-10-23. Some Monotonicity Properties of Convex Functions with Applications. https://doi.org/10.1007/s00009-014-0440-z

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