I have the following constraint
$$\sum_{i=2}^{n}p_i\leq \sum_{i=1}^{n-1}\log(1+ax_i)$$
and
$$p_i \geq 0$$
where $a > 0$ is constant, and $p_i \geq 0$ and $x_i \geq 0$ are the optimization variables. I want to know how this constraint is convex. Any help in this regard will be much appreciated. Thanks in advance.
Since $f(x)=-\log(1+ax)$ is a convex function,
$$\sum_{i=2}^{n}p_i- \sum_{i=1}^{n-1}\log(1+ax_i)$$
is a convex function, since sum of convex function is convex and sum of $p_i$ is clearly a convex function as well.
Hence $$\sum_{i=2}^{n}p_i- \sum_{i=1}^{n-1}\log(1+ax_i) \le 0$$ is a convex set.
The constraints $p_i \geq 0$ is also convex and intersection of convex set is convex.