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 Hermite–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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Rooin, J., Dehghan, H. Some Monotonicity Properties of Convex Functions with Applications. Mediterr. J. Math. 12, 593–604 (2015). https://doi.org/10.1007/s00009-014-0440-z
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DOI: https://doi.org/10.1007/s00009-014-0440-z