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The now canonical proof of Schwarz's Lemma appeared in a 1907 paper ofCarath\'eodory, who attributed it to Erhard Schmidt. Since then, Schwarz'sLemma has acquired considerable fame, with multiple extensions andgeneralizations. Much less known is that, in the same year 1907, Landau andToeplitz obtained a similar result where the diameter of the image set takesover the role of the maximum modulus of the function. We give a new proof ofthis result and extend it to include bounds on the growth of the maximummodulus. We also develop a more general approach in which the size of the imageis estimated in several geometric ways via notions of radius, diameter,perimeter, area, capacity, etc...

Area, capacity and diameter versions of Schwarz's Lemma