Birkhoff–James Orthogonality and the Zeros of an Analytic Function

Abstract

Bounds are obtained for the zeros of an analytic function on a disk in terms of the Taylor coefficients of the function. These results are derived using the notion of Birkhoff–James orthogonality in the sequence space $$\ell ^p$$ with $$p \in (1, \infty )$$ , along with an associated Pythagorean theorem. It is shown that these methods are able to reproduce, and in some cases sharpen, some classical bounds for the roots of a polynomial.