Construction of an inner function in the little Bloch space

Stephenson, Kenneth

doi:10.1090/S0002-9947-1988-0951624-3

Construction of an inner function in the little Bloch space
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Trans. Amer. Math. Soc. 308 (1988), 713-720 Request permission

Abstract:

An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space ${\mathcal {B}_0}$. In addition to showing how an inner function can meet the geometric conditions for ${\mathcal {B}_0}$, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.

References

Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
J. M. Anderson, On division by inner factors, Comment. Math. Helv. 54 (1979), no. 2, 309–317. MR 535061, DOI 10.1007/BF02566274
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
Burgess Davis, Brownian motion and analytic functions, Ann. Probab. 7 (1979), no. 6, 913–932. MR 548889
P. L. Duren, H. S. Shapiro, and A. L. Shields, Singular measures and domains not of Smirnov type, Duke Math. J. 33 (1966), 247–254. MR 199359
José L. Fernández, Singularities of inner functions, Math. Z. 193 (1986), no. 3, 393–396. MR 862885, DOI 10.1007/BF01229806
Renate McLaughlin, Exceptional sets for inner functions, J. London Math. Soc. (2) 4 (1972), 696–700. MR 296309, DOI 10.1112/jlms/s2-4.4.696
George Piranian, Two monotonic, singular, uniformly almost smooth functions, Duke Math. J. 33 (1966), 255–262. MR 199320

Blaschke products in

David A. Stegenga and Kenneth Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2) 24 (1981), no. 2, 243–254. MR 631937, DOI 10.1112/jlms/s2-24.2.243
Kenneth Stephenson, The finite range of an inner function, Bull. London Math. Soc. 11 (1979), no. 3, 300–303. MR 554397, DOI 10.1112/blms/11.3.300
Sakari Toppila, On exceptional values of functions meromorphic outside a linear set, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 115–119. MR 595181, DOI 10.5186/aasfm.1980.0526

Trigonometric series