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1.
一类高阶周期线性微分方程解的性质及复振荡   总被引:3,自引:0,他引:3  
本文对一类超越型高阶周期线性微分方程解的性质及复振荡证明了:设B(ξ)=g1(1/ξ) g2(ξ),其中g1(t)和g2(t)是整函数,以及g1(t)(或g2(t))是超越的且级小于1/2。令A(z)=B(e^z)。(i)如果方程ω^(k) A(z)ω=0(k≥3)有解f(z)≡0满足log^ N(r,1/F)=0(γ),则f(z)和f(z 2πi)线性相关;(ii)如果B(ξ)在ξ=∞(或相应地在ξ≠0)有一p阶极点,p不被整除,则前方程的任一解f(z)≠0的零收敛指数都是无究,且更强的结论log^ N(r,1/f)≠0(γ)成立。  相似文献   

2.
Making use of the linear operator Lp^m(λ,e)f(z)=1/z^p+∞∑k=1[e/e+λk]^m akz^k-p,where e〉0,λ〉0,p∈N,m∈N0=NU{0},z∈U^* and f(z)∈∑p,we introduce two subclasses of meromorphic p-valent analytic functions and investigate convolution and inclusion properties for these classes.  相似文献   

3.
Let B be the unit ball in a complex Banach space. Let S^*k+1(B) be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k + 1 of f(z) - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^*k+1(B).  相似文献   

4.
Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f(z) = a→f′(z) = a, and f′(z) = a →f^(k)(z) = a, then either f = Ce^λz + a or f = Ce^λz + a(λ - 1)/)λ, where C and ), are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.  相似文献   

5.
In this article, the zeros of solutions of differential equation f(k)(z)+A(z)f(z) = 0, (*) are studied, where k 2, A(z) = B(ez), B(ζ) = g1(1/ζ) + g2(ζ), g1 and g2 being entire functions with g2 transcendental and σ(g2) not equal to a positive integer or infinity. It is shown that any linearly independent solutions f1, f2, . . . , fk of Eq.(*) satisfy λe(f1 . . . fk) ≥σ(g2) under the condition that fj(z) and fj(z+ 2πi) (j = 1, . . . , k) are linearly dependent.  相似文献   

6.
In this note, the author introduces some new subcIasses of starlike mappings S^*Ωn1p2,…,pn(β,A,B)={f∈H(Ω):|itanβ+(1-itanβ)2/p(z)аp/аz(z)Jf^-1(z)f(z)-1-AB/1-B^2|〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n:|z1|^2+n∑j=2|zj|^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈(-π/2,π/2).Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F(z)=(f(z1)+f'(z1)Pm(z0),(f'(z1))^1/mz0)'where f is a normalized biholomorphic function on the unit disc D, z = (z1,z0) ∈Ωn1p2,…,pn,z0=(z2,…,zn)∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.  相似文献   

7.
在Qp空间上建立了Jackson型不等式,即对任意f(z)=∑j=0 ^∞ αjz^j∈Qp,0≤p〈∞,α〉1及k-1∈N,有 ||f(z)-Г(k)/Г(k+α)∑ j=0 ^k-1Г(k-j+α)/Г(k-j)αjz^j||Qp≤C(α)ω(1/k,f,Qp),其中ω(1/k,f,Qp)为Qp空间中的连续模,C(α)是仅与参数α有关的正常数.  相似文献   

8.
设A(z)是方程f″+P(z)f=0的非零解,其中P(z)是n次多项式,B(z)是一个超越整函数且满足ρ(B)≤1/2,那么方程f″+Af′+Bf =0的每一个非零解都是无穷级.并且方程f″+A(z)f=0两个线性无关解乘积的零点序列收敛指数为无穷.  相似文献   

9.
An analytic function f in the unit disk D := {z ∈ C : |z| 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k(z) := z/(1-z)2, z ∈ D, if there exists δ ∈ (-π/2,π/2) such that Re {eiδ(1-z)2f′(z)} 〉 0, z ∈ D. For the class C(k) of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.  相似文献   

10.
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 satisfies
4∧β/π∧′=1/σ.
The constant in the above inequality is exact.  相似文献   

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