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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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[1] V.V. Arestov, On integral inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 3–22 (in Russian), English Transl. Math. USSR Izv. 18 (1982), 1–17. [2] A. Aziz and N.A. Rather, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl. 289 (2004), 14–29. [3] A. Aziz and N.A. Rather, A refinement of a theorem of Paul Turan concerning polynomials, Math. Inequal. Appl. 1 (1998), 231—238. [4] A. Aziz and W. M. Shah, Inequalities for a polynomial and its derivative, Math. Ineq. Appl. 7 (2004), 379–391. [5] S. Bernstein, Sur la limitation des derivees des polnomes, C. R. Acad. Sci. Paris 190 (1930), 338–341. [6] N.G. De-Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch. Proc. 50 (1947), 1265–1272. [7] V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Math. 32 (1997), 45–51. [8] P.D. Lax, Proof of a conjecture of P. Erd¨os on the derivative of a polynomial. Bull. Amer. Math. Soc. 50 (1944), 509–513. [9] A. Liman, R.N. Mohapatra, and W.M. Shah, Inequalities for the polar derivative of a polynomial, Complex Anal. Oper. Theory 6 (2012), 1199–1209. [10] S.A. Malik, B.A. Zargar, F.A. Zargar, and F.A. Sofi, Turan type inequalities for a class of polynomials with constraints, Int. J. Nonlinear Anal. Appl. 12 (2021), 583–594. [11] A. Mir and A. Wani, Polynomials with polar derivatives, Funct. Approx. 55 (2016), 139–144. [12] Q. I. Rahman and G. Schmeisser, Lp inequalities for polynomials, J. Approx. Theory, 53 (1998), 26–32. [13] N. A. Rather, N. Wani, T. Bhat, and I. Dar, Inequalities for the generalized polar derivative of a polynomial, Int. J. Nonlinear Anal. Appl. 16 (2025), no. 6, 153—159. [14] X. Zhao, Integral inequality for the polar derivatives of polynomials, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 2, 371–378. [15] A. Zygmund, A remark on conjugate series, Proc. London Math. Soc. 34 (1932), 392–400. | ||
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