Let $\alpha, \beta, \gamma$ be positive constants. I am interested in finding conditions on these such that the function \begin{align*} f: \mathbb{C}^2 &\to \mathbb{C}\\ f(z,w)&= \exp\Big(\alpha\,{\bf Re}(z\overline{w}) - \beta |z|² - \gamma |w|²\Big) \end{align*} is bounded.
I found that $4\beta \gamma \geq \alpha^2$ is sufficient for the boundedness. But is this also necessary?
You can write
$$f(z,w) = \exp \bigl((\alpha - 2\sqrt{\beta\gamma})\operatorname{Re} (z\overline{w}) - \bigl\lvert \sqrt{\beta} z - \sqrt{\gamma}w\bigr\rvert^2\bigr).$$
Therefore
$$f(z,\sqrt{\beta/\gamma}z) = \exp\bigl((\alpha - 2\sqrt{\beta\gamma})\sqrt{\beta/\gamma}\lvert z\rvert^2\bigr),$$
and we see that
$$\alpha \leqslant 2\sqrt{\beta\gamma}$$
is necessary for the boundedness of $f$.