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Lq mean extension for the polar derivative of a polynomial | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 29 اردیبهشت 1404 اصل مقاله (355.3 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2025.37212.5426 | ||
| نویسندگان | ||
| Mahmood Bidkham* 1؛ Ahmad Motamednezhad2 | ||
| 1Department of Mathematics, University of Semnan, Semnan, Iran | ||
| 2Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran | ||
| تاریخ دریافت: 02 فروردین 1404، تاریخ پذیرش: 11 اردیبهشت 1404 | ||
| چکیده | ||
| For a polynomial $p(z)$ of degree $n$, we consider an operator $D_{\alpha}$ which map a polynomial $p(z)$ into $D_{\alpha}p(z):=(\alpha-z)p'(z)+np(z)$ with respect to $\alpha$. It was proved by Liman et al [ A. Liman, R. N. Mohapatra and W. M. Shah, Inequalities for the polar derivative of a polynomial, Complex Anal. Oper. Theory, 2012] that if $p(z)$ has no zeros in $|z|<1$ then for all $\alpha,\ \beta\in \mathbb{C}$ with $|\alpha|\geq 1 , \ |\beta|\leq 1$ and $|z|=1$, \begin{align*} \begin{split} |zD_{\alpha}p(z)+n\beta\frac{|\alpha|-1}{2}&p(z)|\leq \frac{n}{2}\{ [|\alpha+\beta\frac{|\alpha|-1}{2}|+|z+\beta\frac{|\alpha|-1}{2}|] \max_{|z|=1}|p(z)|. \end{split}\end{align*} In this paper, we present the integral $L_q$ mean extension of the above inequality for the polar derivative of polynomials. Our result generalize certain well-known polynomial inequalities. | ||
| کلیدواژه‌ها | ||
| Polynomial؛ Integral inequality؛ Polar derivative؛ Restricted zeros | ||
| مراجع | ||
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