Subordination and $H^ p$ functions
HTML articles powered by AMS MathViewer
- by Rahman Younis PDF
- Proc. Amer. Math. Soc. 110 (1990), 653-660 Request permission
Abstract:
Let $\phi$ and $W$ be inner functions with $\phi (0) = W(0) = 0$. It is shown that if $F$ is an exposed point of the unit ball of ${H^1}$ and \[ F(W({e^{it}}))/F(\phi ({e^{it}})) > 0\] almost everywhere, then $F \circ W = F \circ \phi$. If $f = zF$ such that $F$ and $1/F$ are in ${H^r}$ and ${H^s}$, respectively, where $1/r + 1/s \leq 2$ and $\phi$ is a finite Blaschke product, then a necessary and sufficient condition is provided in order for $f(W({e^{it}}))/f(\phi ({e^{it}}))$ to be positive almost everywhere.References
- Yusuf Abu-Muhanna, $H^1$ subordination and extreme points, Proc. Amer. Math. Soc. 95 (1985), no.ย 2, 247โ251. MR 801332, DOI 10.1090/S0002-9939-1985-0801332-7
- Yusuf Abu-Muhanna and D. J. Hallenbeck, Subordination by univalent $H^1$ functions, Bull. London Math. Soc. 19 (1987), no.ย 3, 249โ252. MR 879512, DOI 10.1112/blms/19.3.249
- P. R. Ahern and D. N. Clark, On functions orthogonal to invariant subspaces, Acta Math. 124 (1970), 191โ204. MR 264385, DOI 10.1007/BF02394571
- Carl C. Cowen, On equivalence of Toeplitz operators, J. Operator Theory 7 (1982), no.ย 1, 167โ172. MR 650201
- Karel de Leeuw and Walter Rudin, Extreme points and extremum problems in $H_{1}$, Pacific J. Math. 8 (1958), 467โ485. MR 98981, DOI 10.2140/pjm.1958.8.467
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- D. J. Hallenbeck and T. H. MacGregor, Linear problems and convexity techniques in geometric function theory, Monographs and Studies in Mathematics, vol. 22, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 768747
- Jyunji Inoue and Takahiko Nakazi, Polynomials of an inner function which are exposed points in $H^1$, Proc. Amer. Math. Soc. 100 (1987), no.ย 3, 454โ456. MR 891144, DOI 10.1090/S0002-9939-1987-0891144-2
- Takahiko Nakazi, Exposed points and extremal problems in $H^{1}$, J. Funct. Anal. 53 (1983), no.ย 3, 224โ230. MR 724027, DOI 10.1016/0022-1236(83)90032-0 โ, The kernel of Toeplitz operators, J. Math. Soc. Japan 38 (1980), 607-616. J. Ryff, Subordination ${H^p}$ functions, Duke Math. J. 33 (1966), 347-354.
- Rahman Younis, Hankel operators and extremal problems in $H^1$, Integral Equations Operator Theory 9 (1986), no.ย 6, 893โ904. MR 866970, DOI 10.1007/BF01202522
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 653-660
- MSC: Primary 30D55; Secondary 30C80
- DOI: https://doi.org/10.1090/S0002-9939-1990-1027103-9
- MathSciNet review: 1027103