Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|...Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.展开更多
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...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.展开更多
基金Supported by the NNSF of China(10571164)Supported by Specialized Research Fund for the Doctoral Program of Higher Education(SRFDP)(2050358052)Supported by the NSF of Zhejiang Province(Y606197)
文摘Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.
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）.
文摘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.
