A new proof of pick's theorem

f3 real. (See, for instance, [1], p. 224, ex. 4.) We offer the following proof whic h, although it also uses Bieberbach's result, is considerably different. PROOF: Let cP (z) J~ . Then cP sends the open unit disk into itself. Let cP 1= cP and inductively 1 a? define cPn=cP OcPlI t. If cPll(Z)=An,1 z+AIl,z Z2+ . .. ,it is clear that A"'=M' A I ,2= M' AII ,I = A", AIl I,t, and A II ,2 = AI ,I A nI,2 + A I,2 A;' _1.1' It follows that An,1 = A;'., and A II ,2 =A I ,zA :' ,-; 1 (l +A 1 , 1 +ALI + . . +A :',-;1). Now cPli/A 11 , 1 E!F for each n, so Bieberbach's theorem implies that IA" ,2/A" ,11 :s;: 2, or