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

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