پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شماره‌ها | 608 |
| تعداد مقالات | 8,999 |
| تعداد مشاهده مقاله | 67,027,351 |
| تعداد دریافت فایل اصل مقاله | 7,616,294 |
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 | ||
| تاریخ دریافت: 08 آذر 1400، تاریخ بازنگری: 12 اردیبهشت 1401، تاریخ پذیرش: 14 اردیبهشت 1401 | ||
| 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 | ||
| مراجع | ||
|
1] A. Aziz, Inequalities for the polar derivative of a polynomial, J. Approx. Theory 54 (1988), 306–313. [2] A. Aziz, Inequalities for the polar derivative of a polynomial, J. Approx. Theory 55 (1988), 183–193. [3] A. Aziz and N.A. Rather, On an inequality concerning the polar derivative, Proc. Indian Acad. Sci. Math. Sci. 117 (2003), 349–357. [4] A. Aziz and N.A. Rather, Some Zygmund type Lp inequalities for polynomials, J. Math. Anal. Appl. 289 (2004), 14–29. [5] A. Aziz and W.M. Shah, Some inequalities for the polar derivative of a polynomial, Proc. Indian Acad. Sci. (Math. Sci.) 107 (1997), no. 3, 263–270. [6] S.N. Bernstein, On best approximation of continuous functions by polynomials of given degree, Soobsc. Harkov. Mat. Ob˘s˘c. (2) 13 (1912), 47–194. [7] K.K. Dewan and A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531. [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] M.A. Malik, On the derivative of a polynomial, J. Lond. Math. Soc. 2 (1969), no. 1, 57–60. [10] M. Marden, Geometry of polynomials, 2nd edition, Math. Surveys, No. 3, Amer. Math. Soc. 1996. [11] A. Mir and S.A. Baba, Some integral inequalities for the polar derivative of a polynomial, Anal. Theory. Appl. 27 (2011), no. 4, 340–350. [12] N.A. Rather, Lp inequalities for the polar derivative of a polynomial, J. Ineq. Pure. Appl. Math. 9 (2008), Article 103. [13] N.A. Rather, S.H. Ahangar and G. Suhail, On the polar derivative of a polynomial, Mat. Vesnik 67 (2015), 56–60. [14] P. Tura´n, Uber die ableitung von polynomen ¨ , Compositio Math. 7 (1939), 89–95. | ||
|
آمار تعداد مشاهده مقاله: 43,888 تعداد دریافت فایل اصل مقاله: 333 |
||