Picard type theorems concerning certain small functions

Let f(z) be a meromorphic function in the complex plane, whose zeros have multiplicity at least k + 1(k ≥ 2). If sin z is a small function with respect to f(z), then f~(k)(z)-P(z) sin z has infinitely many zeros in the complex plane, where P(z) is a nonzero polynomial of deg(P(z)) ≠ 1.     

有限级超越整函数的差分多项式的值分布

研究了差分多项式H(z)=POk∑(i=1)a_if(z+c_i)的值分布,其中f是有限级超越整函数,P(f)是,的多项式,κ≥2,ci(i=1,…,k)是互不相同的常数,α_i(i=1,…,κ)是非零常数.得到了H(z)-a和H(z)-α(z)的零点的个数的估计,其中a∈C且α(z)(■0)为小函数.讨论了H(z)的非零有限Borel例外值的不存在性.     

一类非线性复微分差分方程解的不存在性

该文研究了一类复微分差分方程[f(z)f'(z)]n + fm(z + r) = 1,[f(z)f'(z)]n + [f(z + r)-f(z)]m = 1,[f(z) f'(z)] 2 + P2(z) f2(z + η) = Q(z)eα(z) 的超越整函数解,其中P(z), Q(z)为非零多项式,α(z)为多项式,...     

Spectral density functions of bivariable stable polynomials

The Ramanujan Journal - The relationship between a stable multivariable polynomial p(z) and the Fourier coefficients of its spectral density function $$1/|p(z)|^2$$ , is further investigated. In...     

一类无穷下级整函数的Julia集的径向分布

主要研究方程f"(z)+A(z)f'(z)+B(z)f(z)=0(A(z)),B(z)为整函数)的解、解的多项式或微分多项式这些具有无穷下级的整函数的Julia集的径向分布问题.     

一类微分方程的解和小函数的关系

研究二阶线性微分方程 f'+e^az f'+h(z)e^bz f =0 的解以及它们的一阶、二阶、三阶导数, 微分多项式取小函数的点的收敛指数, 其中a, b 是非零复常数且a =cb(c>1), h(z)是非零多项式.     

多项式的根的一个性质的判定

文献[1]在讨论多项式型的函数迭代方程的局部解析解的存在性时涉及到了多项式的根的一个性质.本文给出了判定该性质是否成立的一个简洁的条件,证明了多项式λ_nzⁿ+…+λ₂z²+λ₁z+λ₀有一个根α满足inf{｜λ_nα^nm+…+λ₂a^2m+λ₁α^m+λ₀｜：m=2,3,…｝>0当且仅当如下两个条件之中至少有一个成立：(i)该多项式有一个根β满足｜β｜>1;(ii)该多项式有一个根β满足｜β｜<1,且λ₀≠0.     

给定零点的最近复系数多项式计算问题

Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f(z) and a zero α, in this paper we explore the problem of computing a complex polynomial f(z) such that f(α) = 0 and the distance ∥f-f ∥ is minimal. Considering most of the existing works focus on either certain polynomial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of f(z), we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f(z) explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.     

On an Inequality of Pual Turan Concerning Polynomials-Ⅱ

Let P(z) be a polynomial of degree n and for any complex number α, let D_αP(z) = nP(z)+(α- z) P′(z) denote the polar derivative of the polynomial P(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.     

Some Sharpening and Generalizations of a Result of T. J. Rivlin

Let p(z)=a_0+a_1z+a_2z~2+a_3z~3+···+a_nz~n be a polynomial of degree n.Rivlin[12]proved that if p(z)≠0 in the unit disk,then for 0r≤1,max|z|=r|p(z)|≥((r+1)/2)~nmax|p(z)||z|=1.In this paper,we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin’s Theorem.     

Algebraic composition operators

The focus of this paper is on the following problem. Given a linear space F of complex-valued functions on a set X and a polynomial p(z), is there an algebraic composition operator on F whose characteristic polynomial equals p(z)? We show that the supply of all the polynomials p(z) for which the answer to this question is affirmative depends heavily on the structure of the space F.     

Some generalizations involving the polar derivative for an inequality of Paul Turán

Let p(z) be a polynomial of degree n and for a complex number α, let D _α p(z) = np(z) + (α-z)p'(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros in |z| ≤ K. Our results generalize and sharpen a famous inequality of Turán and some other known results in this direction.     

关于方程f″+Af′+ Bf=0解的增长性,其中系数A是一个二阶线性微分方程的解

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.     

Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations

Summary. By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation for an matrix X, where is a matrix power series whose coefficients are Toeplitz matrices. The function is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degreeN and . These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors that quadratically converges to the vector having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of given is arithmetic operations, where is the cost of solving an Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. From the numerical experiments performed with several test polynomials and power series, the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial root-finders based on recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown. Received September 9, 1998 / Revised version received November 14, 1999 / Published online February 5, 2001     

On the coefficients of a polynomial with restricted zeros

LetP(Z)=αn Zn + αn-1Zn-1 +…+α0 be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coeffi-cients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).     

On the Rate of Overconvergence of the Generalized Eneström-Kakeya Functional for Polynomials

The classical Eneström-Kakeya Theorem, which provides an upper bound for the moduli of zeros of any polynomial with positive coefficients, has been recently extended by Anderson, Saff and Varga to the case of any complex polynomial having no zeros on the ray [0,$+∞$). Their extension is sharp in the sense that, given such a complex polynomials $p_n(z)$ of degree $n≥1$, a sequence of multiplier polynomial can be found for which the Eneström-Kakeya upper bound, applied to the products $Q_{mi}(z)$ · $p_n(z)$, converges, in the limit as $i$ tends to $∞$, to the maximum of the moduli of the zeros of $p_n(z)$. Here, the rate of convergence of these upper bounds is studied. It is shown that the obtained rate of convergence is best possible.     

A NOTE ON A PROBLEM OF BOAS R.P.

To answer the rest part of the problem of Boas R. P. on derivative of polynomial, it is shown that if $\[p(z)\]$ is a polynomial of degree n such that $\[\mathop {\max }\limits_{\left| z \right| \le 1} \left| {p(z)} \right| \le 1\]$ and $\[{p(z) \ne 0}\]$ in $\[\left| z \right| \le k,0 < k \le 1\]$, then $\[\left| {{p^''}(z)} \right| \le n/(1 + {k^n})\]$ for $\[\left| z \right| \le 1\]$. The above estimate is sharp and the equation holds for $\[p(z) = ({z^n} + {k^n})/(1 + {k^n})\]$.     

BMOA空间的循环元

在文中，刻划了BMOA空间的循环元，得到了f是BMOA空间循环元的充分必要条件是f为外函数．     

微分方程f″+e^{az}f′+Q(z)f=F(z) 的复振荡

该文研究了线性微分方程f″+e^{az}f′+Q(z)f=F(z)的复振荡问题，其中Q(z)、F(z )( 0)是整函数，且σ(Q)=1,σ(F)<+∞,Q(z)=h(z)e^{bz},h(z)是多项式，b≠-1是复常数，那么上述线性微分方程的所有解f(z)满足~λ(f)=λ(f)=σ(f)=∞,~λ_2(f)=λ_2(f)=σ_2(f)=1.至多除去两个例外复数a及一个可能的有穷级例外解f_0(z)。     

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations

In this paper, we investigate the growth of solutions of the differential equations f(k)+Ak?1(z)f(k?1)+· · ·+A0(z)f =0, where Aj(z)(j=0, · · · , k?1) are entire functions. When there exists some coe?cient As(z)(s ∈ {1, · · · , k?1}) being a nonzero solution of f00+P(z)f =0, where P(z) is a polynomial with degree n(≥1) and A0(z) satisfiesσ(A0)≤1/2 or its Taylor expansion is Fabry gap, we obtain that every nonzero solution of such equations is of infinite order.