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My research in theoretical physics led to the necessity of constructing an entire function with modulus decaying in the significant part of the complex plane. I wonder whether this is possible because all my attempts to combine known examples of entire functions show that the result is decaying at most for the half of the complex plane.

Let me put the sharp version of the question. Can the modulus of entire function decay for all directions in the complex plane except the positive real axis? I don't see the obvious contradiction of such a behaviour with Picard's little theorem and Liouville's theorem. However, I failed to construct an example. I would be grateful for references about stronger results constraining the behaviour of entire functions.

For my research I seek an entire function with somewhat weaker condition which decays everywhere except the arguments of complex numbers $z$ satisfying the condition $\Re z > a (\Im z)^2$ with some real positive $a$.

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    $\begingroup$ This question was answered here: mathoverflow.net/questions/190837/… If this is not enough, state your question more precisely. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 9, 2020 at 12:27
  • $\begingroup$ Thank you very much for the reference! Anver Kiro suggested such an example in the explicit form of the Taylor series here: mathoverflow.net/q/191663. However, I am confused with the fact that this series is divergent for |z|>1 if the notation log^n (...) is assumed to be (log(...))^n. But an entire function must be represented by the Tailor series which are convergent everywhere. If I misunderstand the notation I would be grateful to learn what does this mean. $\endgroup$
    – Anna
    Commented Aug 9, 2020 at 22:58
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    $\begingroup$ You are right: that answer with Taylor series makes no sense. The correct one is the first answer. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 10, 2020 at 1:36

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