EQUIVALENCE OF ANALYTIC AND SOBOLEV-POINCARE-INEQUALITIES FOR PLANAR DOMAINS

For a finitely connected planar domain Omega it is shown that the anal ytic-Poincare inequality \\f(z) - f(z(0))\\(Lp(Omega)) less than or eq ual to K-p(a)(Omega)\\f'(z)\\(Lp(Omega)) holds uniformly for all holom orphic functions f on Omega (z(0) is an element of Omega fixed, K-p(a) (Omega) an absolute constant) if and only if the Sobolev-Poincare ineq uality \\u(z)\\(Lp(Omega)) less than or equal to K-p(Omega)\\del u(z)\ \(Lp(Omega)) holds for an absolute constant K-p(Omega) and for all u i s an element of C-1(Omega) whose integral over Omega is zero, This pap er extends a result of Hamilton (1986) who established this equivalenc e when 1 < p < infinity.