Counting real zeros of analytic functions satisfying linear ordinary differential equations (Q1912513)

Let \(K\) be a compact real segment and let \(U\) be a simply-connected open subset of the complex plane containing \(K\). Suppose that \(f\) satisfies a linear differential equation of the form \(a_0 f^{(\nu)}+ a_1 f^{(\nu- 1)}+ \cdots+ a_{\nu- 1} f'+ a_\nu f=0\) where \(f\) and the coefficients \(a_k\) are analytic in \(U\) and continuous in the closure \(\overline {U}\) of \(U\). The first result says that if \(a_0 \equiv 1\), then the number of zeros of \(f\) in \(K\), counted according to multiplicity, does not exceed \(\beta (A+ \nu)\), where \(\beta\) is a constant depending only on the geometry of the pair \((U, K)\) and where \[ A= \max_{k=0, \dots,\nu} \max_{z\in \overline {U}} |a_k (z)|. \] A second result addresses the case where \(a_0\) is non-constant.