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This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.A Dirichlet Norm Inequality and Some Inequalities for Reproducing Kernel Spaces
Jacob Burbea
Proceedings of the American Mathematical Society
Vol. 83, No. 2 (Oct., 1981), pp. 279285
Published by: American Mathematical Society
DOI: 10.2307/2043511
Stable URL: http://www.jstor.org/stable/2043511
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Abstract
Let $f$ be analytic and of finite Dirichlet norm in the unit disk $\Delta$ with $f(0) = 0$. Then, for any $q > 0$, $$\\exp f\^2_q \leqslant \exp\bigg\{\frac{1}{\pi q} \int_\Deltaf'(z)^2 d\sigma(z) \bigg\} (d\sigma(z) \equiv (i/2) dz \bigwedge d\bar z).$$ Equality holds if and only if $f(z) = q \log(1  z\bar{\zeta}$ for some $\zeta \in \Delta$. Here, for $g(z) = \sum^\infty_{n = 0} b_nz^n$, analytic in $\Delta$, $$\g\^2_q \equiv \sum^\infty_{n = 0} \frac{n!}{(q)_n} b_n^2,$$ where $(q)_0 = 1$ 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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Proceedings of the American Mathematical Society © 1981 American Mathematical Society