巫朝霞,劉衛(wèi)霞

(1.新疆財(cái)經(jīng)大學(xué)應(yīng)用數(shù)學(xué)學(xué)院,新疆烏魯木齊 830011;2.中國(guó)人民大學(xué)經(jīng)濟(jì)學(xué)院,北京 100872)

關(guān)于代數(shù)多項(xiàng)式的一個(gè)積分表示式

巫朝霞1,劉衛(wèi)霞2

(1.新疆財(cái)經(jīng)大學(xué)應(yīng)用數(shù)學(xué)學(xué)院,新疆烏魯木齊 830011;2.中國(guó)人民大學(xué)經(jīng)濟(jì)學(xué)院,北京 100872)

研究了代數(shù)多項(xiàng)式導(dǎo)數(shù)的Bernstein不等式和M arkov不等式.通過代數(shù)多項(xiàng)式導(dǎo)數(shù)的一個(gè)積分表示式,給出這兩個(gè)著名不等式以及它們的離散形式的證明.

代數(shù)多項(xiàng)式；Cauchy積分公式；Bernstein不等式；M arkov不等式

1 引言

本文中用Πn表示n次復(fù)系數(shù)代數(shù)多項(xiàng)式全體,用Tn表示n次復(fù)系數(shù)三角多項(xiàng)式全體.用D表示單位圓盤{z:|z|≤1},用?D表示單位圓周{z:|z|=1}.對(duì)于p∈Πn用‖p‖D和‖p‖C[?1,1]分別表示多項(xiàng)式p(z)在D上和區(qū)間[?1,1]上的上確界范數(shù).有如下的著名Bernstein不等式和M arkov不等式:

1941年,文[2]首先提出Markov不等式中的上界不必是|pn(z)|在整個(gè)區(qū)間[?1,1]上的上確界,可以改進(jìn)為|pn(z)|在區(qū)間[?1,1]上的一些離散點(diǎn)上的上確界.這就是

定理DC[2]設(shè)pn(z)∈Πn,那么

其中c是常數(shù),Cn(z)=cosn arccosz,z∈[?1,1],是第一類Chebyshev多項(xiàng)式(參見文[1]).

1985年,文[3]對(duì)Bernstein不等式進(jìn)行了相應(yīng)改進(jìn),得到如下不等式:

定理FRR[3]設(shè)pn(z)∈Πn,那么

2 幾個(gè)引理

2 定理的證明及其推論

參考文獻(xiàn)

[1]Lorentz G G.App roxim ation of functions[M].Austin:Holt,Rinehart and W inston,1966.

[2]Duffin R J,Schaeffer A C.A refinement of an inequality of the brothers Markoff[J].Trans.Amer.Math. Soc.,1941,50:517-528.

[3]Frappier C,Rahman Q I,Ruscheweyh St.New inequalities for polynomials[J].Amer.Math.Soc.,1985,288: 69-99.

[4]D ryannov D,Fournier R.On a discrete variant of Bernstein’s polynomial inequality[J].Analysis,2005,25:73-77.

[5]K rylov V I.App roxim ate Calcu lation of Integrals[M].New York:M acm illan,1962.

[6]Dryannov D.Quad rature formulaewith free nodes for periodic functions[J].Numer.Math.,1994,67:441-464.

A integral rep resen tation of algeb raic polynomials

WU Zhao-xia1,LIUWei-xia2

(1.School of App lied Mathem atics,X in jiang University of Finance and Econom ics,U rm q,830011 China; 2.School of Econom ics,Renmin University of China,Beijing,100872 China)

In thispaper,we discuss the Bernstein’s inequality and Markov’s inequality for algebraic polynomials and its derivative.By using the integral representation for polynomial and its derivative,we give another proof of the discrete Bernstein’s inequality and asimilar outcome of Markov’s inequality.

algebraic polynomials,Cauchy integral formula,Bernstein’s inequality,M arkov’s inequality

O174.42

A

1008-5513(2009)02-0325-07

2008-06-10.

國(guó)家自然科學(xué)基金(10671155).

巫朝霞(1975-),碩士,講師,研究方向：主要從事高等數(shù)學(xué)的教學(xué)與研究.

2000M SC:41A 10,41A 25