# Does there exist a study of entire functions which satisfy $|F(x+iy)| \leq a e^{-bx^2}e^{cy^2}$?

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.

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 $$c it follows that $$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}$$.