# Entire $f(z)$ such that $0 < | f(a_+ + b i)| < \exp(-a_+^4)$?

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.

• If find the $a_+$ notation confusing. Do you mean $| f(a + b i)| < \exp(-a^4)$ for real $a > 0$ and all real $b$? Aug 19 '21 at 14:50
• @MartinR YES. exactly. positive real or real $> 0$ is the same. I will edit though.
– mick
Aug 19 '21 at 18:43
• I considered ideas like $f(z+1)=f(z)^2$ but the usual solutions to this equation ( $c^{2^z}$ ) do not work.
– mick
Aug 19 '21 at 18:54

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 $$c \to \infty$$