3 added 330 characters in body

The exponential function grows in module as the exponential of the real part. Therefore, the set of all $z$ such that $|exp(az)|\leq \sup_K |exp(a\times\cdot)|$ is a half space containing $K$, and meeting $K$. You get all such half spaces, if you vary $a$ in $\mathbb{C}$ or even on the unit circle. Their intersection is the convex hull of $K$ by some famous theorem on convex sets (Krein-Milman?).

So by restricting yourself to the exponential functions you get the convex hull of $K$. The hull you're interested in is a subset of that set.

I don't know what $K^*$ stands for, but it won't be the convex hull in general. for instance, if you take $\Omega=\mathbb{C}\setminus\lbrace 0\rbrace$ and $K=$ the unit circle, and $f(z)=z, ~g(z)=\frac{1}{z}$, you see that the hull you're interested in is just $K$ itself. The convex hull may not even be a subset of $\Omega$.

2 deleted 24 characters in body; added 17 characters in body

The convex hull of $K$ is the intersection of semispaces. The exponential function grows in module as the exponential of the real part. The Therefore, the set of all $z$ such that $|exp(az)|\leq \sup_K |exp(a\times\cdot)|$ is a half space containing $K$, and meeting $K$. You get all such half spaces, if you vary $a$ in $\mathbb{C}$, and their \mathbb{C}$or even on the unit circle. Their intersection is the convex hull of$K$by some famous theorem on convex sets (Krein-Milman?). So by restricting yourself to the exponential functions you get the convex hull of$K$. The hull you're interested in is a subset of that set. 1 The convex hull of$K$is the intersection of semispaces. The exponential function grows in module as the exponential of the real part. The set of all$z$such that$|exp(az)|\leq \sup_K |exp(a\times\cdot)|$is a half space containing$K$. You get all such half spaces, if you vary$a$in$\mathbb{C}$, and their intersection is the convex hull of$K$by some famous theorem on convex sets (Krein-Milman?). So by restricting yourself to the exponential functions you get the convex hull of$K\$. The hull you're interested in is a subset of that set.