Some Inequalities for the Polynomial with S-Fold Zeros at the Origin

Let p（z） be a polynomial of degree n, which has no zeros in ｜z｜〈 1, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, pp. 38-41] established｜zp＇（z）＋nβ/2p（z）｜≤n/2{（｜β/2｜＋｜1＋β/2｜）｜z｜=1max｜p（z）｜-（｜1＋β/2｜-｜β/2｜）｜z｜=1min｜p（z）｜},for any ｜β｜≤ 1 and ｜z｜ = 1. In this paper we improve the above inequality for the polynomial which has no zeros in ｜z｜ 〈 k, k≥ 1, except s-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.

Some Integral Mean Estimates for Polynomials with Restricted Zeros

Let P（z） be a polynomial of degree n having all its zeros in ｜z｜≤ k. For k = 1, it is known that for each r 〉 0 and ｜a｜≥1,n（｜a｜-1）{∫0^2π｜P（e^iθ）｜^rdθ}^1/r≤{∫0^2π｜1＋e^iθ｜^rdθ}^1/rmax｜z｜=1｜DzP（z）｜.In this paper, we shall first consider the case when k≥1 and present certain generaliza- tions of this inequality. Also for k≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.

Some Inequalities on Polar Derivative of Polynomial Having No Zero in a Disc

Let , , be a polynomial of degree n having no zero in , , then Qazi [Proc. Amer. Math. Soc., 115 (1992), 337-343] proved . In this paper, we first extend the above inequality to polar derivative of a polynomial. Further, as an application of our result, we extend a result due to Dewan et al. [Southeast Asian Bull. Math., 27 (2003), 591-597] to polar derivative.

Polar Derivative Versions of Polynomial Inequalities

Let be a polynomial of degree n and for a complex number , let ?denote the polar derivative of the polynomial ?with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.

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.

ON EXTREMAL PROPERTIES FOR THE POLAR DERIVATIVE OF POLYNOMIALS

If p（z） is a polynomial of degree n having all its zeros on ｜z｜ = k, k ≤ 1, then it is proved[5] that In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.

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 andAppl. Math., 9 （2008）, Issue 4, Art. 103] that for every γ 〉 0 and every real or complex number a with ｜ α ｜ ≥ 1,{∫ 2π 0｜DαP（e^iθ）｜γdθ｜}^1/γ≤n（｜α｜＋1）Cγ{∫2π0｜P（e^iθ）｜γ^dθ}^1/γ,Cγ={1/2π∫2π 0｜1＋e^iβ｜^γdβ}^-1/γ,where DaP（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 Results on the Polar Derivative of a Polynomial

Let P（z） be a polynomial of degree n and for any complex number α, let DαP（z） = nP（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.

Some Lγ Inequalities for the Polar Derivative of a Polynomial

In this paper, we consider an operator Da which maps a polynomial P（z） in to DaP（z）：= np（z）＋ （a-z）P＇（z）, where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Our results yields several generaliza- tions 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.

Inequalities for the Polar Derivatives of a Polynomial

Let P（z） be a polynomial of degree n, having all its zeros in ｜z｜≤ 1. In 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.

Inequalities for the Polar Derivative of a Polynomial

If is a polynomial of degree , having all its zeros in |z|≤K, K≥1 , then it was proved by Aziz and Rather [2] that for every real or complex number with |a| ≥K, . In this paper, we sharpen above result for the polynomials p(z) of degree n>

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;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 zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.

Jacobi多项式及其导数零点的求解方法及算法实现

建立Jacobi多项式及其任意阶导数零点求解方法的统一框架。并在该框架下给出了算法和程序。数值例子表明该方法是非常有效的。

New Inequalities on L_γ-Spaces

We consider for a fixed μ, the class of polynomialsPn,μ,s：＋{P（z）=z^2（anz^n-s＋∑j=μ^n-s an-jz^n-j-s）;1≤μ≤n-s}of degree n, having all zeros in ｜ z｜ ≤ k, k≤ 1, with s-fold zeros at the origin. In this paper, we have obtained inequalities in the reverse direction for the above class of polynomials. Besides, extensions of some Turan-type inequalities for the polar derivative of polynomials have been considered.

L^q Inequalities and Operator Preserving Inequalities

Let Pn be the class of polynomials of degree at most n. Rather and Shah [15] proved that if P∈Pn and P（z） 6=0 in｜z｜〈1, then for every R〉0 and 0≤q〈∞,｜B[P（Rz）]｜q≤｜RnB[zn]＋λ0｜q｜1＋zn｜q ｜P（z）｜q, where B is a Bn-operator. In this paper, we prove some generalization of this result which in particular yield-s some known polynomial inequalities as special. We also consider an operator Dαwhich maps a polynomial P（z） into DαP（z）：=nP（z）＋（α-z）P′（z） and obtain exten-sions and generalizations of a number of well-known Lq inequalities.

Chebyshev多项式及其插值法在函数求导中的应用

为了更精确地用插值函数的导数逼近复杂函数的导数,本文基于Chebyshev多项式和最小零偏差定理提出了一种应用Lagrange插值求复杂函数导数的新方法。我们首先以n+1次Chebyshev多项式的零点作为插值节点进行Lagrange插值,进而用插值函数的导数值逼近被插值函数的导数值。误差分析和数值算例表明本文所提出的方法在复杂函数求导中取得了良好的效果。

多项式函数零点重数的一个性质

研究探讨了多项式函数零点和零点的重数与函数D_k(f(x))=f^((k))(x)/f^((k+1))(x)之间的关系,得出了相应的结论.