In this paper, we establish some new inequalities in the plane that are inspired by some classical Bernstein-type inequalities that relate the sup-norm of a polynomial to that of its derivative on the unit circle. The obtained results relate the \(L^\gamma \)-norm of the polar derivative and the polynomial. Our results besides derive polar derivative analogues of some classical Bernstein-type inequalities also include several interesting generalizations and refinements of some integral-norm inequalities for polynomials, as well.

中文翻译：

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一类多项式II的伯恩斯坦型积分不等式

在本文中，我们在平面上建立了一些新的不等式，这些不等式是由一些经典的伯恩斯坦型不等式启发的，这些不等式将多项式的乘范数与其在单位圆上的导数的乘范数联系起来。获得的结果与极数导数的\（L ^ \ gamma \）-范数和多项式相关。我们的结果除了得出一些经典的伯恩斯坦型不等式的极导数类似物外，还包括多项有趣的概括和多项式的某些积分范数不等式的细化。