1. Suppose that the functions x=x(a, 3), y=y(a, f) define a oneto-one harmonic mapping of the unit disc P in the a, p3-plane (a+i3 ==y) onto a convex domain C in the x, y-plane (x+iy=z). The origin is assumed to be fixed. Introducing two functions F(y) and G(y) which, in r, depend analytically upon the variable y we may write z = Re F(,y) +i Re G(y). The purpose of the present paper is (i) to give a new proof of a lemma which, in a special form, was first used by T. Rado [13] and which was proved in general by L. Bers (see [2, Lemma 3.3]),2 (ii) to derive an improved value for an important constant first introduced by E. Heinz [3]. The proofs will be very simple due to the fact that there is a close connection between univalent harmonic mappings and the minimal surface equation (see e.g. [11, footnote 2]) and also the differential equation