Suppose we have two numbers $m,n\in \mathbb{N}$ with $n>4$ and $n$ even. I am looking for a function $f: \{1,\dots,m\} \to \mathbb{R}^+$ with the following properties:
- $f(k+1)<f(k)$ for all $k\in \{1,\dots,m-1\}$
- $\frac{f(1)}{f(m)}< 1 + \frac{1}{\frac{n}{2}-1}$
- $\frac{f(1)-f(k-1)}{f(k-1)-f(k)}< \frac{1}{\frac{n}{2}-1}$ for all $k\in \{2,\dots,m-1\}$
It seems easy to find such a function, but I am stuck. Can you help?