Smoothness of inverse Laplace transforms of functions univalent in a half-plane (Q1061892)

\textit{W. K. Hayman} [Ann. Acad. Sci. Fenn., Ser. A I 250/13, 12 p. (1958; Zbl 0102.065)] introduced the class of functions f univalent in Re z\(>0\) which satisfy \(\limsup_{x\to \infty}x^ 2| f(x)| \leq 1.\) He proved that such an f has a representation of the form \[ f(z)=\int^{\infty}_{0}a(t)e^{-tz}dt \] for some function a(t). \textit{D. H. Hamilton} [Trans. Am. Math. Soc. 276, 323-333 (1983; Zbl 0535.30018)] proved that a is of class \(C^ 1\). Here we improve on Hamilton's result by showing that in fact \(a\in C^{1,\alpha}\) for some \(\alpha >\). Our technique uses a conformally invariant localization of an integral estimate of Clunie and Pommerenke. It has been used also in another paper [Complex Variables (to appear; Zbl 0553.30013)] to prove that \[ M(r,f)=O(1-e)^{-\alpha}\Rightarrow a_ n=O(n^{\alpha - 1})\quad for\quad \alpha \geq -1/320, \] where now f denotes a function univalent in \(| z| <1\).