Let $f:[1, +\infty)\rightarrow\mathbb{R}$ be a function such that $\displaystyle{\lim_{n\to\infty}}f(n) = 0$, and put $f_{\sigma}(x) := \sum_{k=1}^\infty f(k) - f(k+x)$.
Note that the above series is convergent if $f$ is monotone, convex or concave (see https://www.ijmex.com/index.php/ijmex/article/view/60).
Now, we are looking for a such function with the following properties:
(a) $f$ is concave and strictly increasing,
(b) $f_{\sigma}(x-1)\geq F(x) - F(1)$,
where, $F$ is a primitive function of $f$ (so $F(x) - F(1) = \int_{1}^x f(t) dt$),
(c) The function $f_{\sigma}(x) - \frac{1}{2}f(x) - F(x)$ is convex on $[1, +\infty)$.
Note. One can show that the above conditions imply $f_{\sigma}(x) - \frac{1}{2}f(x)$ and $F(x)$ are both convex. I tried and I found some examples, such as $f(x) = -x^r$ for $r<-1$ which it satisfied the conditions (a), (b) but it does not satisfy the condition (c).
Thanks in advance.