臺灣博碩士論文加值系統

In 1991, Korenblum presented his conjecture on
Bergman spaces. He speculated that if $f(z)$ and $g(z)$ are two
holomorphic functions on the unit disc in the complex plane, then
there exists a number $0<c<1$ such that the condition
"$|f(z)|\geq|g(z)|$" could implies "$||f||_{2}\geq||g||_{2}$", where
$||.||_{2}$ is the Bergman norm. This maximum principle was
confirmed in 1999 by Hayman. It is not only an analogous property
with $H^{p}$ spaces but also inspires numerous research in other
function spaces. In this article we introduce the development of
this problem and the related research results from which the idea
originally came from Korenblum's maximum principle. In the end we
give some discussions about the circumstances when the functions are
constrained in the form of $ rod(z-a_{i})$ for $ -1 \leq a_{i} \leq
1$.

1. Introduction ...2
2. Main results ...10
References ...15

1. B. Korenblum, A maximum
principle for the Bergman space, Publ. Mat., 35, (1991), 479-486.

2. B. Korenblum, and K. Richards, Majorization and Domination in the Bergman Space, Proc. Amer.
Math. Soc., 117, (1993), 153-158.

3. B. Korenblum, R. O'Neil, K. Richards and K. Zhu, Totally Monotone Functions with Applications to the
Bergman Space, Trans. Amer. Math. Soc., 337, (1993), 795-806.

4. J. Matero, On Korenblum's
maximum principle for the Bergman space, Proc. Arch. Math.
, 64, (1995), 337-340.

5. W. Schwick, On Korenblum's
maximum principle, Proc. Amer. Math. Soc., 125, (1997), 2581-2587.

6. N. Danikas and W.K. Hayman, Domination on Sets and in $H^{p}$, Results Math., 34, (1998), 85-90.

7. W. K. Hayman, On a conjecture of
Korenblum, Analysis, 19, (1999), 195-205.

8. A. Hinkkanen, On a maximum principle in Bergman space, J. Anal. Math., 79, (1999), 335-344.

9. A. Schuster, The maximum principle for the Bergman space and the Mobius pseudodistance for the annulus, Proc. Amer. Math. Soc., 134, (2006), 3525-3530.

10. C. Wang, Domination in the Bergman Space and Korenblum’s Constant, Integral Equations and
Operator Theory, 61, (2008), 423--432.

11. C. Wang, Behavior of the
constant in Korenblum’s maximum principle, Math. Nachr., 281, (2008), 447-454.

12. S.-C., Chen, On dominating sets
for uniform algebra on pseudoconvex domains, Journal of Pure
and Applied Mathematics Quarterly, special issue in honor of J. J. Kohn, 6, no. 3, (2010), 715-724.

13. C. Wang, Some results on Korenblum's maximum principle, 373, (2011), J. Math. Anal. Appl., 393-398.

14. S.-C., Chen, On dominating sets for Nevanlinna class (I), Taiwanese J. Math., 15, no.
4, (2011), 1829-1840 .