Integral inequality for the polar derivatives of polynomials

	Integral inequality for the polar derivatives of polynomials
International Journal of Nonlinear Analysis and Applications
مقاله 34، دوره 13، شماره 2، مهر 2022، صفحه 371-378 اصل مقاله (332.77 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.25391.3001
نویسنده
Xingjun Zhao^*
School of Mathematics, Renmin University of China, Beijing, 100872, China
چکیده
Let $P(z)$ be a polynomial of degree $n$ and for any complex number $\alpha$, let $$D_{\alpha}P(z)= nP(z) + (\alpha - z)P^{\prime}(z)$$ denote the polar derivative of $P(z)$ with respect to a complex number $\alpha$. In this paper, we prove some $L_{r}$ inequalities for the polar derivative of a polynomial have all zeros in $\|z\| \leq 1$. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, {\it Inequalities for the polar derivative of a polynomial}, J. Interd. Math. 10 (2007), no. 4, 525--531] and includes as special cases several interesting many known results.
کلیدواژه‌ها
Polar derivative؛ Inequality of polynomials؛ Integral inequality؛ Zeros
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