Computer investigation of Landau’s theorem

Abstract:

Let $f(z) = {a_0} + {a_1}z + \cdots$ be regular for $\left | z \right | < 1$ and never take the values $0$ and $1$; then $\left | {{a_1}} \right |$ has a bound depending only on ${a_0}$. J. A. Jenkins gave an explicit bound (Canad. J. Math. 8 (1956), 423–425) $\left | {{a_1}} \right | \leqq 2\left | {{a_0}} \right |\left \{ {\left | {\log } \right |\left . {{a_0}} \right \| + 5.94} \right \}$. The author investigates the shapes for the curves $\left | {{a_1}} \right | \leqq L{\text {(}}{a_0}{\text {)}}$ for given ${a_0}$ by the aid of a computer and shows that although Jenkins’ result is about right when ${a_0}$ is negative, 4.38 will be the best possible constant in his form and that a much smaller estimate should be available when ${a_0}$ is positive or complex.