The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions.
The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation.
Conversely if Un converges to a kernel not equal to C, then by the Koebe quarter theorem Un contains the disk of radius f 'n(0) / 4 with centre 0.
By the Koebe distortion theorem Hence the sequence fn is uniformly bounded on compact sets.
Uniqueness in the Riemann mapping theorem forces f = g, so the original sequence fn is uniformly convergent on compact sets.