On Jack's lemma

Abstract

We give a new proof and minor refinements, in the polynomial case, of Jack's lemma. This proof is essentially based on the Bernstein's inequality for polynomials on the unit circle. We also discuss the cases of equality of \[ \frac {d^2}{d \theta ^2} \ln | {f(e^{i\theta })}| \leq 0, \] where $f$ is analytic in the unit disc and in a neighborhood of $e^{i\theta }$ where $|{f(e^{i\theta })}|=\sup _{|z|\lt 1}|f(z)|$.