Length and area estimates of the derivatives of bounded holomorphic functions

Abstract:

MacGregor [1] and Yamashita [5] proved the estimates of the coefficient ${a_n}$ of the Taylor expansion $f(z) = {a_0} + {a_n}{z^n} + \cdots$ of $f$ nonconstant and holomorphic in $|z| < 1$ in terms of the area of the image of $|z| < r < 1$ by $f$ and the length of its outer or exact outer boundary. We shall consider some analogous estimates in terms of the non-Euclidean geometry for $f$ bounded, $|f| < 1$, in $|z| < 1$. For example, $2\pi {r^n}|{a_n}|/(1 - |{a_0}{|^2})$ is strictly less than the non-Euclidean length of the boundary of the image of $|z| < r$, the multiplicity not being counted.