On the operator ranges of analytic functions

Author:
J. S. Hwang

Journal:
Proc. Amer. Math. Soc. **87** (1983), 90-94

MSC:
Primary 47A60; Secondary 30C45

DOI:
https://doi.org/10.1090/S0002-9939-1983-0677239-3

MathSciNet review:
677239

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Abstract | References | Similar Articles | Additional Information

Abstract: Following Doob, we say that a function analytic in the unit disk has the property if and for some arc on the unit circle whose measure ,

**1**]. Finally, we prove that the operator range of any -fold univalent function is not convex. This extends our solution to Ky Fan's Problem [

**4**].

**[1]**J. L. Doob,*The ranges of analytic functions*, Ann. of Math. (2)**36**(1935), 117-126. MR**1503212****[2]**-,*Conformally invariant cluster value theory*, Illinois J. Math.**5**(1961), 521-549. MR**0186821 (32:4276)****[3]**N. Dunford and J. T. Schwartz,*Linear operators, Part*I:*General theory*, Interscience, New York, 1958. MR**1009162 (90g:47001a)****[4]**Ky Fan,*Analytic functions of a proper contraction*, Math. Z.**160**(1978), 275-290. MR**0482310 (58:2383)****[5]**E. Hille,*Analytic function theory*. II, Ginn, Boston, Mass., 1962. MR**0201608 (34:1490)****[6]**J. S. Hwang and D. C. Rung,*Proof of a conjecture of Doob*, Proc. Amer. Math. Soc.**75**(1979), 231-234. MR**532142 (81i:30061)****[7]**-,*An improved estimate for the Bloch norm of functions in Doob's class*, Proc. Amer. Math. Soc.**80**(1980), 406-410. MR**580994 (81i:30058)****[8]**J. S. Hwang,*On an extremal property of Doob's class*, Trans. Amer. Math. Soc.**252**(1979), 393-398. MR**534128 (80i:30057)****[9]**-,*On the ranges of analytic functions*, Trans. Amer. Math. Soc.**260**(1980), 623-629. MR**574804 (81g:30039)****[10]**-,*On estimate for Bloch and Doob norm and covering problem in Doob's class*, J. Math. Anal. Appl. (to appear).**[11]**-,*A problem on Riesz-Dunford operator and convex univalent function*, Glasgow Math. J. (to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0677239-3

Keywords:
Analytic function,
Riesz-Dunford operator,
convex function and operator range

Article copyright:
© Copyright 1983
American Mathematical Society