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...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.展开更多
In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈...In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈 R. Our result generalizes and refines some known polynomial inequalities.展开更多
基金supported by UGC under major research project scheme vide No. MRP-MAJOR-MATH-2013-29143
文摘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.
文摘In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈 R. Our result generalizes and refines some known polynomial inequalities.
