ON CERTAIN INEQUALITIES FOR CONVEX FUNCTIONS

Abstract

In this note, we show that when the function $f(x)$ is convex in the interval $(0, \infty)$, for any natural number $n$,
$$
\sum_{k=1}^n f(k) \leq \frac{n(f(1)+f(n))}{2}
$$
is true. Using this, we give upper bounds for $\sum_{k=1}^n k^k$ and $\frac{a^n-1}{a-1}$ $(a>1).$

Received: January 3, 2023  
Revised: January 31, 2023