A natural boundary for solutions to the second order pantograph equation (Q703871)

The authors study solutions to functional differential equations of the form \[ y(z)-ay'(z)-by(z)+\lambda y(az)=0, \] where \(\alpha,a\) and \(b\) are real constants such that \(\alpha>1\), \(a>0\), and \(b>0\). Here, \(z\) is a complex variable and \(\lambda\) is a nonzero parameter. They prove that if a solution \(y\) is holomorphic in some connected open subset of \(\{z\in\mathbb{C}: \text{Re}\,z>0\}\) whose boundary contains the imaginary axis and \(y\) has a singularity at \(z=0\), then \(y\) has the imaginary axis as its natural boundary.