Some Results on the Polar Derivative of a Polynomial

Let P(z) be a polynomial of degree n and for any complex number α, let Dα P(z) = n P(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to α. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.     

Inequalities for the Polar Derivatives of a Polynomial

Let P(z) be a polynomial of degree n, having all its zeros in|z| ≤1In this paper, we estimate kth polar derivative of P(z) on|z|= 1 and thereby obtain compact generalizations of some known results which among other things yields a refinement of a result due to Paul Tura'n.     

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

Some L~γ Inequalities for the Polar Derivative of a Polynomial

In this paper,we consider an operator D_α which maps a polynomial P(z)in to D_αP(z):=np(z) +(α-z)P'(z),where α∈■ and obtain some L~γ inequalities for lucanary polynomials having zeros in |z|≤k≤1.Our results yields several generalizations and refinements of many known results and also provide an alternative proof of a result due to Dewan et al.[7],which is independent of Laguerre's theorem.     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

Derivative Estimates of Iterated Spherical Average and Extension

In the paper, we study derivative estimates of the iterated spherical averages(A_t)~N(f).We obtain the optimal range of exponents(α, N, p) to ensure the Lpboundedness of P(?/(?_x))(A_1)~N(f)for 1 ≤ p ≤∞, where P is a homogeneous polynomial of degree α. The main theorem extends some known results. As an application, we obtain the smallest N such that(A_1)~N: L~p(R~n) → L_α~p(R~n).     

Some Inequalities Concerning the Polar Derivative of a Polynomial-II

In this paper, we consider the class of polynomials P（z）= anz^n＋ ∑vn=μan-vz^n-v,1≤μ≤n , having all zeros in ｜z｜≤k, k ≤1 and thereby present an alternative proof, independent of Laguerre＇s theorem, of an inequality concerning the polar derivative of a polynomial.     

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.     

A NOTE ON A PROBLEM OF BOAS R.P.

To answer the rest part of the problem of Boas R. P. on derivative of poiyaomial, it is shown that if p(s) is a polynomial of degree n such that ■|p(z)|≤1 and. p(s)≠0 in |z|≤k, 0相似文献   

L~p INEQUALITIES AND ADMISSIBLE OPERATOR FOR POLYNOMIALS

Let p(z) be a polynomial of degree at most n. In this paper we obtain some new results about the dependence of p(Rz)-βp(rz) + α (R+1/r+1)n-|β | p(rz) s on p(z) s for every α, β∈ C with |α|≤ 1, |β | ≤ 1, R > r 1, and s > 0. Our results not only generalize some well known inequalities, but also are variety of interesting results deduced from them by a fairly uniform procedure.     

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.     

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).     

Some inequalities for the polar derivative of a polynomial

LetP(z) be a polynomial of degreen which does not vanish in |z|<1. In this paper, we estimate the maximum and minimum moduli of thekth polar derivative ofP(z) on |z|=1 and thereby obtain compact generalizations of some known results, which among other results, yields interesting refinements of Erdos-Lax theorem and a theorem of Ankeny and Rivlin.     

Extensions of some polynomial inequalities to the polar derivative

Let p(z) be a polynomial of degree n and for any real or complex number α, let Dαp(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 inside or outside a circle. Our results shall generalize and sharpen some well-known polynomial inequalities.     

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+f^m(z+η)=1,[f(z)f′(z)]n+[f(z+η)?f(z)]^m=1,[f(z)f′(z)]^2+P^2(z)f^2(z+η)=Q(z)e^α(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,m,n为正整数,η∈C?{0},并给出了这类方程不存在超越整函数解的几个充分条件.     

Bank-Laine functions with periodic zero-sequences

A Bank-Laine function is an entire function E such that E(z) = 0 implies that E’(z) = ±1. Such functions arise as the product of linearly independent solutions of a second order linear differential equation ω″ + A(z)ω = 0 with A entire. Suppose that $$E(z)=R(z)e^{g(z)}\prod_{j=1}^m \prod_{k=1}^{q_j}(e^{\alpha_jz}-\beta_{j,k}),$$ where R is a rational function, g is a polynomial, and the α_j and β_j,k are non-zero complex numbers, and that E’(z) = ±1 at all but finally many zeros z of E. Then the quotients α_j/α_j′ are all rational numbers and E is a Bank-Laine function and reduces to the form E(z) = P₀ (e^αz) e^Q ₀(z) with α a non-zero complex number and P₀ and Q₀ polynomials.     

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 $\[p(z) = ({z^n} + {k^n})/(1 + {k^n})\]$.