 By F. van Oystaeyen, A. Verschoren

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Sample text

23. Let 0 < p < ∞. An analytic function f (z) on the disc D is in the Hardy space H p if 1 − R2 = ∞ sup 0

If u ∗α ∈ L p (∂D), then u is the Poisson integral of some f ∈ L p (∂D). See Appendix A for the corresponding results when p < 1. 12. A compact set K is locally connected if whenever U is a relatively open subset of K and z ∈ U ⊂ K there is a relatively open subset V of K such that V is connected and z ∈ V ⊂ U. Let be a simply connected domain such that ∂ contains at least two points. Prove ∂ is locally connected if and only if the Riemann map ϕ : D → extends continuously to D. Hint: For one direction, use the uniform continuity of ϕ.

Again let ϕ be univalent and assume ϕ(z) = 0 for all z. 6) where C4 does not depend on ϕ. Hayman  attributes the following argument to P. L. Duren: By Exercise 16, {|z|