Sharp Weak‐Type Inequalities for Analytic Functions on the unit Disc

If Q is a Steiner symmetric domain, symmetric about the y ‐axis, then it can be proved that for an analytic function F : D → C ( D = {zɛ C : |z| < 1}), such that Im F (0) = 0, and for t > 0, the inequality| { z ɛ T : F ( z ) ∉ t ċ Q } | ⩽ C Q ċ ∥ f ∥ 1 tholds, where‖ f ‖ 1 = ∫ ‐ π π| f ( e t v) | d v , | A |denotes the Lebesgue measure of a set A ⊂ T and C Q is some constant dependent only on Q . The best constant C Q is found. Some weak‐type inequalities follow, which generalize results of Baernstein.