This comes from Hörmander's "An Introduction to Complex Analysis in Several Variables".

We defined the $A(\Omega)$-hull (analytic functions in an open set $\Omega$). $\hat{K}$ of a compact set $K\subset\Omega$ by $\hat{K}=\{z;z\in\Omega, |f(z)|\leq\sup_K |f| \operatorname{for every } f\in A(\Omega) \}$.

The book says, if we consider $f(z)=e^{az}$ for every complex number $a$, we obtain $\hat{K}\subseteq \operatorname{convex hull of }K$.

I do not get how he concluded this result. I do not know how to turn a $\hat{z}\in\hat{K}$ into a linear combination of elements $z\in K$ using the exp function. Are there specific $a$ I need to choose? Can I construct this?

Also, he says "Furthermore, it is clear that $\overset{*}{K}=\hat{K}$ ". I'm assuming that the $K$ with the weird star mark on top represents the convex hull? Even then, I do not understand the reverse inclusion.