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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. 279-285
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.
A Dirichlet Norm Inequality and Some Inequalities for Reproducing Kernel Spaces
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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 $q > 0$, $$\|\exp f\|^2_q \leqslant \exp\bigg\{\frac{1}{\pi q} \int_\Delta|f'(z)|^2 d\sigma(z) \bigg\} (d\sigma(z) \equiv (i/2) dz \bigwedge d\bar z).$$ 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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