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?
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1$\begingroup$ What do you exactly mean by "a good sequence"? As for the question in the title, the following thread in MO might be relevant: mathoverflow.net/a/29735/1593 $\endgroup$ – José Hdz. Stgo. Jan 23 at 21:01

1$\begingroup$ A similar question was asked here mathoverflow.net/questions/27100/… $\endgroup$ – Alexandre Eremenko Jan 24 at 15:32

1$\begingroup$ Also see mathoverflow.net/questions/321550/… $\endgroup$ – Alexandre Eremenko Jan 24 at 15:57