A Pointwise Bound for a Holomorphic Function which is Square-Integrable with Respect to an Exponential Density Function

Let $\phi$ be a real-valued smooth function on $\mathbf{C}$ satisfying $0 \le \Delta \phi \le M$ for some $M \ge 0$. We consider the space of all holomorphic functions which are square-integrable with respect to the measure $e^{-\phi(z)} dz$. In this paper, a pointwise bound for any function in this space is established. We show that there exists a constant $K$ depending only on $M$ such that $|f(z)|^2 \le Ke^{\phi(z)}\|f\|^2$ for any $f$ in this space and for any complex number $z$.