If $f(z)$ is an entire function (analytic in the complex plane), with the following property:
There exist $r_0>0$ such that $$|f(re^{it})|\leq g(r)$$ for all $r>r_0$, and all $t\in [0,2\pi]$ ($g(r)$ is some continuous function of $r$, for all $r>0$).
How I can show that: Given $0<r\leq r_0$ there exists $M_0>0$ such that $$|f(re^{it})|\leq M_0g(r).$$
Edit: $g(r)=e^{cr}, c>0$.