Write $\log(\cdot) \le -\exp(\cdot)$ in the DCP form

Question

I am interested in writing $$ \log \left(\exp(-\sqrt x)+ \left(1 + \frac{y}{z} \right)^{y} \right) \leq -\exp(-z), \qquad x\geq0,y\geq0,z>0 $$ in the Disciplined Convex Programming (DCP) form. I don't know how to convert this expression to a DCP form. I am learning Convex Optimization by self-study. I use CVXPY to implement expressions.

Answer 1

I think I get the solution.

It could be written as

log_sum_exp(-sqrt(x), kl_div(y + z, z) + kl_div(z, y + z)) <= -exp(-z),

where log_sum_exp(x_1,x_2) means $\log\left(\exp(x_1)+\exp(x_2)\right)$, which is convex and kl_div(x,y) means $x\log(x/y)-x+y$, which is also convex.

By simple calculation, we have kl_div(y + z, z) + kl_div(z, y + z)$=y\log(1+\frac{y}{z})$.

Put these back to the original function, we get the answer.