The spherical metric and univalent harmonic mappings

Let f=h+g¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=h+\overline{g}$$\end{document} be a harmonic univalent map in the unit disk D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document}, where h and g are analytic. This paper finds an improved estimate for the second coefficient of h. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25, 1984), and by Sheil-Small (J Lond Math Soc 42:237–248, 1990). When the sup-norm of the dilatation is less than 1, it is also shown that the spherical area of the covering surface of h is dominated by the spherical area of the covering surface of f.