Let p（z） = a0＋a1z＋a2z^2＋a3z3＋…＋anz^n be a polynomial of degree n. Rivlin [12] proved that if p （z） ≠ 0 in the unit disk, then for 0 〈 r ≤ 1,max（｜r＋1｜/2）^n max｜p（z）｜.｜z｜=1In this paper, we prove...Let p（z） = a0＋a1z＋a2z^2＋a3z3＋…＋anz^n be a polynomial of degree n. Rivlin [12] proved that if p （z） ≠ 0 in the unit disk, then for 0 〈 r ≤ 1,max（｜r＋1｜/2）^n max｜p（z）｜.｜z｜=1In 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.展开更多
Let p(z)=be a polynomial degree n and let Then accord-ing to Bernstein's inequality ||p'||<n||p||.It is a well known open problem to obtain inequality analogous to Bernstein's inequality for the class I...Let p(z)=be a polynomial degree n and let Then accord-ing to Bernstein's inequality ||p'||<n||p||.It is a well known open problem to obtain inequality analogous to Bernstein's inequality for the class IIn of polynomials satisfying p(z)≡znp(1/z)Here we obtain an inequality analogous to Bernstein's inequality for a subclass of IIn Our results include several of the known results as special cases.展开更多
Let p(z)=∑v^n=0avz^v anzn be a polynomial of degree n,M(p,R)=:max|z|=R≥0|p(z)|and M(p,1)=:||P||.Then according to a well-known result of Ankeny and Rivlin[1],we have for R≥1,M(p,R≤(R^n+1/2)||p||.This inequality ha...Let p(z)=∑v^n=0avz^v anzn be a polynomial of degree n,M(p,R)=:max|z|=R≥0|p(z)|and M(p,1)=:||P||.Then according to a well-known result of Ankeny and Rivlin[1],we have for R≥1,M(p,R≤(R^n+1/2)||p||.This inequality has been sharpened by Govil[4],who proved that for R≥1,M(p,R)≤(R^N+1/2)||p||-n/2(||p||^2-4|an|^2/||p||){(R-1)||p||/||p||+2|an|-ln(1+(R-1)||p||/||p||+2|an|)}.In this paper,we sharpen the above inequality of Govil[4],which in turn sharpens the inequality of Ankeny and Rivlin[1].展开更多
文摘Let p（z） = a0＋a1z＋a2z^2＋a3z3＋…＋anz^n be a polynomial of degree n. Rivlin [12] proved that if p （z） ≠ 0 in the unit disk, then for 0 〈 r ≤ 1,max（｜r＋1｜/2）^n max｜p（z）｜.｜z｜=1In 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.
文摘Let p(z)=be a polynomial degree n and let Then accord-ing to Bernstein's inequality ||p'||<n||p||.It is a well known open problem to obtain inequality analogous to Bernstein's inequality for the class IIn of polynomials satisfying p(z)≡znp(1/z)Here we obtain an inequality analogous to Bernstein's inequality for a subclass of IIn Our results include several of the known results as special cases.
文摘Let p(z)=∑v^n=0avz^v anzn be a polynomial of degree n,M(p,R)=:max|z|=R≥0|p(z)|and M(p,1)=:||P||.Then according to a well-known result of Ankeny and Rivlin[1],we have for R≥1,M(p,R≤(R^n+1/2)||p||.This inequality has been sharpened by Govil[4],who proved that for R≥1,M(p,R)≤(R^N+1/2)||p||-n/2(||p||^2-4|an|^2/||p||){(R-1)||p||/||p||+2|an|-ln(1+(R-1)||p||/||p||+2|an|)}.In this paper,we sharpen the above inequality of Govil[4],which in turn sharpens the inequality of Ankeny and Rivlin[1].
