Does there exist a non-constant entire function of finite order having zeros of multiplicity at least $k+1,$ where $k$ is a positive integer, such that it omits at least one non-zero value in $\mathbb{C}?$
I am trying to construct an example but the condition on zeros is making it hard to check. Of course, there are infinitely many entire functions omitting a non-zero value in $\mathbb{C}$ when the condition of multiple zeros is dropped.
Such a function does not exist.
An entire function of finite order which omits a non-zero value $c$ is necessarily of the form $f(z) = c + e^{p(z)}$ with a polynomial $p$. (This follows for example from the Hadamard factorization theorem.)
Then $f$ has infinitely many zeros, but $f'(z) = p'(z) e^{p(z)}$ has only finitely many zeros, so almost all zeros of $f$ are simple.