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.
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 $c \to \infty$