Form and properties of the canonical Weierstrass product of an entire function with real values on \(\mathbb{R} \) (Q2306664)

This paper concerns entire functions \(f:\mathbb{C}\rightarrow \mathbb{C}\) with \(f(0)=1\) and \(f(x) \in \mathbb{R}\) for \( x \in \mathbb{R}\), having no real zeros but infinitely many complex conjugate pairs \((z_k,\overline{z_k})\) of zeros with \(|z_k| \le |z_{k+1}|\). It is shown that, if the function \(f\) has order one, then \[ f(z)=e^{f'(0)}W(z), \] where \(W\) is the canonical Weierstrass product \[ W(z)=\prod_{k=1}^\infty \left(1-\frac{z}{z_k}\right)\left(1-\frac{z}{\overline{z_k}}\right)\exp\left(\frac{z}{z_k}+\frac{z}{\overline{z_k}}\right). \] This result is used to specify the proof of Theorem 1 of [Ukr. Math. J. 71, No. 4, 643--650 (2019); translation from Ukr. Mat. Zh. 71, No. 4, 564--570 (2019; Zbl 1436.30024)].