On bounds for the derivative of analytic functions

Abstract:

Let $g(z)$ be analytic and $|g(z)| \leqq 1$ in $|z| < 1;g(z) = \sum \nolimits _{k = p}^\infty {{a_k}{z^k},p \geqq 1}$, then a sharp upper bound is derived for $|g’(z)|$. Let $h(z)$ be analytic for $|z| < 1,h(0) = 1,\operatorname {Re} h(z) > \alpha$ where $0 \leqq \alpha < 1$, then bounds for $|h’(z)|$ are derived and sharpened for a function with missing terms.