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Sharp Weak‐Type Inequalities for Analytic Functions on the unit Disc
Author(s) -
Tomaszewski Boguslaw
Publication year - 1986
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/18.4.355
Subject(s) - mathematics , lebesgue measure , type (biology) , combinatorics , lebesgue integration , domain (mathematical analysis) , measure (data warehouse) , unit (ring theory) , constant (computer programming) , function (biology) , mathematical analysis , ecology , mathematics education , database , evolutionary biology , computer science , biology , programming language
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.