The spherical metric and univalent harmonic mappings

Let f = h + g ¯ be a harmonic univalent map in the unit disk D , 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 .