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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Dirichlet norm inequality and some inequalities for reproducing kernel spaces


Author: Jacob Burbea
Journal: Proc. Amer. Math. Soc. 83 (1981), 279-285
MSC: Primary 30C40; Secondary 30H05, 46E20
MathSciNet review: 624914
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Abstract: Let $ f$ be analytic and of finite Dirichlet norm in the unit disk $ \Delta $ with $ f(0) = 0$. Then, for any $ q > 0$,

$\displaystyle \left\Vert {\exp f} \right\Vert _q^2 \leqslant \exp \left\{ {\fra... ...} }^2}d\sigma (z)} \right\}\quad (d\sigma (z) \equiv (i/2)\;dz \wedge d\bar z).$

Equality holds if and only if $ f(z) = - q\log (1 - z\bar \zeta )$ for some $ \zeta \in \Delta $. Here, for $ g(z) = \Sigma _{n = 0}^\infty {b_n}{z^n}$, analytic in $ \Delta $,

$\displaystyle \left\Vert g \right\Vert _q^2 \equiv \sum\limits_{n = 0}^\infty {\frac{{n!}} {{{{(q)}_n}}}{{\left\vert {{b_n}} \right\vert}^2},} $

where $ {(q)_0} = 1\;{\text{and}}\;{(q)_n} = q(q + 1) \cdots (q + n - 1)$ for $ n \geqslant 1$. This also extends with a substantially easier proof, a result of Saitoh concerning the case of $ q \geqslant 1$. In addition, a sharp norm inequality, valid for two functional Hilbert spaces whose reproducing kernels are related via an entire function with positive coefficients, is established.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0624914-0
PII: S 0002-9939(1981)0624914-0
Keywords: Dirichlet norm, reproducing kernel spaces
Article copyright: © Copyright 1981 American Mathematical Society