On Functions Analytic in an Annulus |
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2M+n lie a₂ ac Schaffer Analytic Functions annulus arguement az/R boundary components boundary function conformally equivalent covering surface DOCTOR OF PHILOSOPHY doubly connected region Duke Mathematical Journal essential singularities finite connectivity following result function H(z,a function with maximal FUNCTIONS ANALYTIC functions f(z functions regular Gronwall Hadamard's three circle harmonic conjugate harmonic measure interval 1,-R Jacobi's identity 2n-1 Let f(z log z./log mapped maximal derivative maximum modulus n+1 coefficients negative axis noted that H(z,a ع obtain H(z,R positive number Proof R²™z rational of degree readily verifed real for real real values greater region is conformally regular and single result follows ROBERT CLELAND CARSON Robinson satisfy Schwarz's lemma sense that H(z sharp bounds simple and 2M+n simple pole simple zeros Single Valued Functions sup f(z Theorem 2.4 three circle theorem unit circle University of Wisconsin unless g(z value w>1 exactly W₁