TY - JOUR

T1 - Mapping properties of analytic functions on the unit disk

AU - Solynin, Alexander Yu

PY - 2008/2

Y1 - 2008/2

N2 - Let f be analytic on the unit disk D with f (0) = 0. In 1989, D. Marshall conjectured the existence of the universal constant ro > 0such that f (roD) C Dm := {w : |w| <M} whenever the area, counting multiplicity, of a portion of f (D)over DM is <πM2. Recently, P. Poggi-Corradini (2007) proved this conjecture with an unspecified constant by the method of extremal metrics. In this note we show that such a universal constant ro exists for a much larger class consisting of analytic functions omitting two values of a certain doubly-sheeted Riemann surface. We also find a numerical value, ro = .03949 ., which is sharp for the problem in this larger class but is not sharp for Marshall's problem.

AB - Let f be analytic on the unit disk D with f (0) = 0. In 1989, D. Marshall conjectured the existence of the universal constant ro > 0such that f (roD) C Dm := {w : |w| <M} whenever the area, counting multiplicity, of a portion of f (D)over DM is <πM2. Recently, P. Poggi-Corradini (2007) proved this conjecture with an unspecified constant by the method of extremal metrics. In this note we show that such a universal constant ro exists for a much larger class consisting of analytic functions omitting two values of a certain doubly-sheeted Riemann surface. We also find a numerical value, ro = .03949 ., which is sharp for the problem in this larger class but is not sharp for Marshall's problem.

KW - Analytic function

KW - Growth theorem

KW - Hyperbolic metric

UR - http://www.scopus.com/inward/record.url?scp=77950603572&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-07-09080-6

DO - 10.1090/S0002-9939-07-09080-6

M3 - Article

AN - SCOPUS:77950603572

VL - 136

SP - 577

EP - 585

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -