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Let $f(z)$ be a nonconstant entire function. Let $a_+$ denote a positive real number; $0 < a_+ $. Let $b$ be a real number.

Is there an entire $f(z)$ such that for any $a_+,b$ we have

$$0 < | f(a_+ + b i)| < \exp(-a_+^4) $$

or even the weaker

$$| f(a_+ + b i)| < \exp(-a_+^4) $$

??

I considered generalizations of Gamma, Barnes G and hypergeometric but I was not able to find a solution.

I know that the domain is called a half-plane and I read about it, but without success.

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    $\begingroup$ If find the $a_+$ notation confusing. Do you mean $| f(a + b i)| < \exp(-a^4)$ for real $a > 0$ and all real $b$? $\endgroup$
    – Martin R
    Aug 19 '21 at 14:50
  • $\begingroup$ @MartinR YES. exactly. positive real or real $> 0$ is the same. I will edit though. $\endgroup$
    – mick
    Aug 19 '21 at 18:43
  • $\begingroup$ I considered ideas like $f(z+1)=f(z)^2$ but the usual solutions to this equation ( $c^{2^z}$ ) do not work. $\endgroup$
    – mick
    Aug 19 '21 at 18:54
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In general if $f$ is analytic in a neighborhood of $\Re z \ge 0$ st $|f(ix)| \le M, |f(z)| \le Ae^{B|z|}, \Re z \ge 0$ and we also have $\lim_{r \to \infty}\frac{\log |f(r)|}{r}=-\infty$, then $f$ is identically zero, which immediately shows that no $f$ as required in the post above can exist since by continuity $|f(ib)| \le 1$ and more generally $|f(z)| \le 1, \Re z \ge 0$ while clearly $\log |f(a)| < -a^4, a>0$ so one has $\lim_{a \to \infty}\frac{\log |f(a)|}{a}=-\infty$

(note that for any positive continuous function $\eta(x)>0, x \in \mathbb R$, one can find an entire (and with no zeroes if one so wishes) $f$ st $0 < |f(x)| < \eta(x), x \in \mathbb R$, but those functions are necessarily big somewhere in the half-plane $\Re z \ge 0$ if $\eta$ is small enough like here where $\eta(x)=\exp(-x^4)$)

The proof is an easy application of one of the generic Phragmen Lindelof theorems to the function $f_c(z)=f(z)e^{cz}, c>0$ which satisfies that $|f_c(ib)| \le M, |f_c(z)| \le Ae^{(B+c)|z|}, \Re z \ge 0$ and $\limsup_{a \to \infty}\frac{\log |f_c(a)|}{a} \le 0$, conditions that imply that $|f_c(z)| \le M, \Re z \ge 0$ which in turn clearly lead to $f(z)$ identically zero by letting $c \to \infty$

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