I would like to prove convexity (verified in Matlab) of a certain function I am working with. Let $\mathcal{B}$ be the set $$\mathcal{B}=\left\{\mathbf{x}=[x(1),\ldots ,x(N)] \; \bigg|\; \sum x(n) = P,\;\; x(n)> 0, \forall n\right\},$$ i.e., the set of all $N$-vectors with positive entries and given sum.
Define the function $$y(\mathbf{x})= x(1) +\sum_{n=2}^N x(n) \,g\left(\sum_{m=1}^{n-1} x(m)\right)$$ where $g(x)$ has the following properties (I have many different variants of $g(x)$, all of which satisfies the below given properties):
- $g(0)=1$
- $g(\infty)=0$
- $g(x)$ is strictly monotonically decreasing
- $g(x)$ is convex
I can verify (Matlab) that the function $y(\mathbf{x})$ is convex over $\mathcal{B}$, i.e., for $\mathbf{x}_1,\mathbf{x}_2 \in \mathcal{B}$ it holds that $$\lambda y(\mathbf{x}_1)+(1-\lambda)y(\mathbf{x}_2)\geq y(\lambda\mathbf{x}_1+(1-\lambda)\mathbf{x}_2).$$
I would like to add that if one removes the constraint that all vectors in $\mathcal{B}$ has given sum, the function is no longer convex, i.e., the function $y(\mathbf{x})$ is not convex over $\mathbb{R}^N$.
If it would turn out that the 4 properties are not sufficient to prove convexity, I provide an example of a $g(x)$ function that I am working with: $$g(x) =\frac{A}{A+x}.$$ for an arbitrary constant $A$.