The gamma function is the only meromorphic function $f$ on $\mathbb{C}$ satisfying $zf(z)=f(z+1)$, $f(1)=1$ and which is logarithmically convex on the positive real axis.
Does there exist a non-meromorphic function $f$ on $\mathbb{C}$ satisfying $zf(z)=f(z+1)$, $f(1)=1$ and which is logarithmically convex on the postitive real axis?
Suppose that $f: \mathbb C \to \mathbb C$ is a function such that $zf(z)=f(z+1)$ and $f(1)=1.$ Then we get
$$1=f(0+1)=0 \cdot f(0)=0.$$
Conclusion ?