Hermitian forms in function theory

Abstract:

Let $f$ and $g$ be analytic in the unit disk $|z|\; < 1$. We give a new derivation of the positive semidefinite Hermitian form equivalent to $|g(z)| \leq |f(z)|$, for $| z | < 1$, and use it to derive Hermitian forms for various classes of univalent functions. Sharp coefficient bounds for these classes are obtained from the Hermitian forms. We find the specific functions required to make the Hermitian forms equal to zero.