An Inequality of Bohr Type on Hardy-Sobolev Classes
Received:March 12, 2007  Revised:May 26, 2007
Key Word: Hardy-Sobolev classes   the spectrum of a function   an inequality of Bohr type.
Fund ProjectL:the National Natural Science Special-Purpose Foundation of China (No.10826079); the National Natural Science Foundation of China (No.10671019); the Initial Research Fund of China Agricultural University (No.2006061).
 Author Name Affiliation LI Xue Hua College of Science, China Agricultural University, Beijing 100083, China
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Let $\beta>0$ and $S_{\beta}:=\{z\in {\mathbb C}:|{\rm Im}\,z|<\beta\}$ be a strip in the complex plane. For an integer $r\geq 0$, let $H_{\infty,\beta}^{r}$ denote those real-valued functions $f$ on ${\mathbb R}$, which are analytic in $S_{\beta}$ and satisfy the restriction $|f^{(r)}(z)|\leq 1$, $z\in S_{\beta}$. For $\sigma>0$, denote by $B_{\sigma}$ the class of functions $f$ which have spectra in $(-2\pi\sigma, 2\pi\sigma)$. And let $B_{\sigma}^{\perp}$ be the class of functions $f$ which have no spectrum in $(-2\pi\sigma, 2\pi\sigma)$. We prove an inequality of Bohr type $$\|f\|_{\infty}\leq\frac{\pi}{\sqrt{\lambda}\Lambda\sigma^r}\sum_{k=0}^{\infty}\frac{(-1)^{k(r 1)}}{(2k 1)^r\sinh((2k 1)2\sigma\beta)}\,, \qquad f\in H_{\infty,\beta}^{r}\cap B_{\sigma}^{\perp},$$ where $\lambda\in (0, 1)$, $\Lambda$ and $\Lambda'$ are the complete elliptic integrals of the first kind for the moduli $\lambda$ and $\lambda'=\sqrt{1-\lambda^2}$, respectively, and $\lambda$ satisfies $$\frac{4\Lambda\beta}{\pi\Lambda'}=\frac{1}{\sigma}.$$ The constant in the above inequality is exact.