Dominating $|f|$ on a compact convex set

Question

I have the following statemant:

Let $K\subset \mathbb{C}$ compact and convex set. Now, for any $z\in\mathbb{C}\setminus K,$ there is $f$ holomorfic on $\mathbb{C}$ such that $||f||_K<|f(z)|.$

where $||f||_K=\text{max}_{w\in K}|f(w)|$ and I was told that I can prove this directly, without Runge's theorem but i have no starting point. I have somehow to construct $f$ depending on $z$...

Answer 1

Hint: Suppose $K\subset \{x+iy: x\le 0\}$ and $z=1.$ Can you do it in this case?