锐化多项式的图兰式不等式

AbstractFor the polynomial \(P(z) = \sum\nolimits_{j = 0}^n {{c}_{j}}{{z}^{j}}\) of degree n having its all zeros in \({\text{|}}z{text{|}} \leqslant k\), \(k \geqslant 1\), Jain in "On the derivative of a polynomial," Bull.Math.Soc.Roumanie Tome 59, 339-347 (2016) proved that\(\mathop {\max }\limits_{|z| = 1}.{\text{|}}P'(z){\text{|}}\ungeqslant n\left( {\frac{{{\text{|}}{{c}_{0}}{\text{|}}+ \,{\text{|}}{{c}_{n}}{text{|}}{{k}^{n + 1}}}}}{{{{\text{|}}{{c}_{0}}{{text{|}}(1 + {{k}^{n + 1}}}) + \,{\text{|}}{{c}_{n}}{\text{|}}({{k}^{{n + 1}}} + {{k}^{2n}}})}}}}\right)\mathop {\max }\limits_{|z| = 1}{在本文中，我们将加强上述不等式和其他相关结果，用于度 \(n \geqslant 2\) 的多项式。