I recently successfully extended a certain result where I use analytic functions which satisfy the following property: $F: \mathbb C \to \mathbb C$ is entire and there exist constants $a,b,c>0$ such that $$ |F(x+iy)| \leq a e^{-bx^2}e^{cy^2} $$ for every $x,y \in \mathbb R$. I was wondering if functions with this property are studied somewhere in the literature. In particular do these functions have maybe a special name? Certainly they are of order less or equal than 2.

## 1 Answer

The inequality in the post can be rewritten as $$\left|F(z)e^{\frac{b+c}{2}z^2}\right|\leq ae^{\frac{c-b}{2}|z|^2},\qquad z\in\mathbb{C}.$$ For $c<b$ it follows that $F=0$, and for $c=b$ it follows that $F(z)$ is a constant times $e^{-\frac{b+c}{2}z^2}$. Focusing on the remaining case $c>b$, we see that studying the functions in the post is essentially the same as studying the entire functions $G:\mathbb{C}\to\mathbb{C}$ satisfying $|G(z)|\leq e^{|z|^2}$ for all $z\in\mathbb{C}$.