Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures

Let \({\mathcal {R}}\) be a polynomially bounded o-minimal expansion of the real field. Let f(z) be a transcendental entire function of finite order \(\rho \) and type \(\sigma \in [0,\infty ]\). The main purpose of this paper is to show that if (\(\rho <1\)) or (\(\rho =1\) and \(\sigma =0\)), the restriction of f(z) to the real axis is not definable in \({\mathcal {R}}\). Furthermore, we give a generalization of this result for any \(\rho \in [0,\infty )\).