Inequalities for a Polynomial and its Derivative

Proceedings Mathematical Sciences

V K Jain¹ 

[1] Mathematics Department, Indian Institute of Technology, Kharagpur 0, India$$

关键词: Inequalities;    zeros;    polynomial.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

For an arbitrary entire function 𝑓 and any 𝑟 > 0, let $M(f, r):=max_{|z|=r}|f(z)|$. It is known that if 𝑝 is a polynomial of degree 𝑛 having no zeros in the open unit disc, and $m:=min_{|z|=1}|p(z)|$, then$$M(p',1)≤frac{n}{2}{M(p,1)-m},$$$$M(p, R)≤left(frac{R^n+1}{2}ight)M(p, 1)-mleft(frac{R^n-1}{2}ight), R>> 1.$$It is also known that if 𝑝 has all its zeros in the closed unit disc, then$$M(p', 1)≥frac{n}{2}{M(p, 1)+m}.$$The present paper contains certain generalizations of these inequalities

【 授权许可】

Unknown