I have been reading this article A characterization of multiplicative linear functionals in Banach algebras and got stuck in the middle of the proof of theorem 1.2 on page 217. In the 3rd line from below, they say that the function $f_{a,b}:\mathbb{C}\longrightarrow\mathbb{C}$ is Lipschitz and entire hence it is affine. Can anyone tell me why it would be affine. Or suggest me a reference to the result which states that an Lipschitz entire complex function will be affine.
Or can you tell other conditions for an entire function to be affine?
Since $f_{a,b}$ is Lipschitz, its derivative is bounded. A bounded entire constant is constant, so $f_{a,b}$ is an antiderivative of a constant, so it is affine.