A problem on the Bloch norm of functions in Doob’s class
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- by J. S. Hwang PDF
- Proc. Amer. Math. Soc. 95 (1985), 554-556 Request permission
Abstract:
Let $\Delta$ denote the unit disc and $\partial \Delta$ denote the unit circle both in the complex plane. Define the Doob’s class $D(\rho )$, $0 < \rho < 2\pi$, as all holomorphic functions on $\Delta$ satisfying (1) $f(0) = 0$, and (2) for some arc ${\Gamma _f} \subseteq \partial \Delta$ with arclength $\rho$, for all $p \in \Gamma$, ${\underline {\lim } _{z \to p}}|f(z)| \geq 1$. Recently the author and Rung [6] proved a conjecture of Doob made in 1935 by showing that the norm \[ ||f|| = {\sup _{z \in \Delta }}(1 - |z{|^2})|f’(z)| \geq \frac {{2\sin \theta (\rho )}}{{e\theta (\rho )}},\quad 0 \leq \theta (\rho ) \leq \pi - \rho /2.\] We then conjecture that the result should be true if the arc ${\Gamma _f}$ is replaced by a finite union of arcs whose total length is at least $\rho$. In this paper, we answer this problem. It turns out to be surprising that the answer depends on the connectivity of the union, namely, the answer is no for the disconnected case, but yes for the connected one.References
- E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999, DOI 10.1017/CBO9780511566134
- Joseph L. Doob, The ranges of analytic functions, Ann. of Math. (2) 36 (1935), no. 1, 117–126. MR 1503212, DOI 10.2307/1968668
- Stephen Dragosh and Donald C. Rung, Normal functions bounded on arcs and a proof of the Gross cluster-value theorem, Hiroshima Math. J. 9 (1979), no. 2, 303–312. MR 535513
- Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
- J. S. Hwang and D. C. Rung, Proof of a conjecture of Doob, Proc. Amer. Math. Soc. 75 (1979), no. 2, 231–234. MR 532142, DOI 10.1090/S0002-9939-1979-0532142-3
- J. S. Hwang and D. C. Rung, An improved estimate for the Bloch norm of functions in Doob’s class, Proc. Amer. Math. Soc. 80 (1980), no. 3, 406–410. MR 580994, DOI 10.1090/S0002-9939-1980-0580994-1
- J. S. Hwang, On an extremal property of Doob’s class, Trans. Amer. Math. Soc. 252 (1979), 393–398. MR 534128, DOI 10.1090/S0002-9947-1979-0534128-6
- Wladimir Seidel, On the distribution of values of bounded analytic functions, Trans. Amer. Math. Soc. 36 (1934), no. 1, 201–226. MR 1501738, DOI 10.1090/S0002-9947-1934-1501738-9
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 554-556
- MSC: Primary 30C45; Secondary 30C80
- DOI: https://doi.org/10.1090/S0002-9939-1985-0810162-1
- MathSciNet review: 810162