Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way.
Suppose $f(z)$ is an entire function s.t. $f(z)=O(e^{c_1|z|^2})$ for some $c_1>0$, and for $x$ real $f(x)=O(e^{-c_2|x|^2})$ for some $c_2>0$. Then $f(x+iy)=O(e^{-ax^2+by^2})$ for some $a,b>0$.