Inequality between $(1, \frac{\alpha}{2})$-Holder norms of two functions

I have to prove that, given $$f\in C^{1, \frac{\alpha}{2}}([a, b])$$, then there exists a constant $$K$$ such that $$\|e^f\|_{1, \frac{\alpha}{2}}\leq K\|f\|_{1, \frac{\alpha}{2}}.$$ I do not undestand how to prove it. Can someone help me?

Thank You