# When is entire function bounded on a ray?

Let $$f(z)=\sum_{n=1}^\infty c_n z^n$$ be entire function on the complex plane. May we express the property $$\int_0^\infty |f(x)|^2 /x dx<\infty$$ or some other property controlling the behavior for large positive $$x$$ in terms of coefficients? This property is quite rare:if we change finitely many coefficients arbitrarily, it no longer holds. To be more specific: for given sequence $$(a_n)$$, can we find all good sequences $$(c_n)$$ for which $$(a_n c_n)$$ is also good? Generically, how many such pairs of good sequences do we have?