A Relative of the Lemma of Schwarz*

so that d(r, 0; ƒ') is the length of the segment on the w-plane between the image of the point 3 = 0 and the image of the point z = re. The lemma of Schwarz is the following: THEOREM 1. Let w=f(z) be analytic f or \z\ < 1 . If d(r,6;f)S 1 for all (r, 6) with r<l, then (1) d(r,0;f)gr and (2) | / ( 0 ) | ^ 1 . The sign of equality holds in (1) (for r^O) and in (2), if and only if \f(z) | = 1 ; that is, if and only if the transformation w=f(z) is a rigid motion. If the (real) function g(z) is subharmonic for \z\ < 1 , then the Lebesgue integral