An Inequality of Bohr Type on Hardy-Sobolev Classes  

作　　者：LI Xue Hua 

机构地区：[1]College of Science, China Agricultural University, Beijing 100083, China

出　　处：《Journal of Mathematical Research and Exposition》2009年第2期213-218,共6页数学研究与评论（英文版）

基　　金：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）.

摘　　要：Let β 〉 0 and Sβ ：= {z ∈ C ： ｜Imz｜ 〈β} be a strip in the complex plane. For an integer r ≥ 0, let H∞^Г,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction ｜f^（r）（z）｜ ≤ 1, z ∈ Sβ. For σ 〉 0, denote by Bσ the class of functions f which have spectra in （-2πσ, 2πσ）. And let Bσ^⊥ be the class of functions f which have no spectrum in （-2πσ, 2πσ）. We prove an inequality of Bohr type‖f‖∞≤π/√λ∧σ^r∑k=0^∞（-1）^k（r＋1）/（2k＋1）^rsinh（（2k＋1）2σβ）,f∈H∞^r,β∩B1/σ,where λ∈（0,1）,∧and ∧′are the complete elliptic integrals of the first kind for the moduli λ and λ′=√1- λ^2,respectively,and λ satisfies4∧β/π∧′=1/σ.The constant in the above inequality is exact.Let β > 0 and Sβ := {z ∈ C : |Imz| < β} be a strip in the complex plane. For an integer r ≥ 0, let H∞r,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction |f(r)(z)| ≤ 1, z ∈ Sβ. For σ > 0, denote by Bσ the class of functions f which have spectra in (2πσ,2πσ). And let Bσ⊥ be the class of functions f which have no spectrum in (2πσ,2πσ). We prove an inequality of Bohr type f ∞≤√πλΛσr∞ k=0 (1)k(r+1) (2k + 1)r sinh((2k + 1)2σβ) , f ∈ H∞r,β∩ Bσ⊥ , where λ∈ (0,1), Λ and Λ are the complete elliptic integrals of the first kind for the moduli λ and λ = √1 λ2, respectively, and λ satisfies 4ΛβπΛ = σ1. The constant in the above inequality is exact.

关 键 词：Hardy-Sobolev classes the spectrum of a function an inequality of Bohr type. 

分 类 号：O174.5[理学—数学]