My research in theoretical physics led to the nessesarity to construct an entire function with modulus decaying in the sighnificant 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 the references about stronger results constraining the behaviour of entire functions.

For my research I seek for 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.