Pairs of paths and critical points (Q5946265)

In this interesting paper under review, the authors establish two sufficient conditions for a holomorphic function \(f(z)\) to have a critical point, \(f'(z_0)=0\), at \(z_0\), where \(z_0\) is a point in the domain of \(f(z)\). These conditions are expressed in terms of the values of \(f\) on a pair of smooth paths which intersect at \(z_0\), where the tangent vectors at \(z_0\) are linearly independent over \(\mathbb R\).