| Title:
|
$L_{p}$ inequalities for the growth of polynomials with restricted zeros (English) |
| Author:
|
Rather, Nisar A. |
| Author:
|
Gulzar, Suhail |
| Author:
|
Bhat, Aijaz A. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
159-167 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $P(z)=\sum _{\nu =0}^{n}a_{\nu }z^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $|z|<1$, then for $1\le p<\infty $ and $R>1$, Boas and Rahman proved \[\left\Vert P(Rz)\right\Vert _{p}\le \big (\left\Vert R^{n}+z\right\Vert_{p}/\left\Vert1+z\right\Vert_{p}\big )\left\Vert P\right\Vert _{p}.\] In this paper, we improve the above inequality for $0\le p < \infty $ by involving some of the coefficients of the polynomial $P(z)$. Analogous result for the class of polynomials $P(z)$ having no zero in $|z|>1$ is also given. (English) |
| Keyword:
|
polynomials |
| Keyword:
|
integral inequalities |
| Keyword:
|
complex domain |
| MSC:
|
26D10 |
| MSC:
|
30C15 |
| MSC:
|
41A17 |
| idZBL:
|
Zbl 07584087 |
| idMR:
|
MR4483050 |
| DOI:
|
10.5817/AM2022-3-159 |
| . |
| Date available:
|
2022-09-01T10:17:56Z |
| Last updated:
|
2023-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150661 |
| . |
| Reference:
|
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| Reference:
|
[2] Arestov, V.: On integral inequalities for trigonometric polynomials and their derivatives.Math. USSR-Izv 18 (1982), 1–17. MR 0607574, 10.1070/IM1982v018n01ABEH001375 |
| Reference:
|
[3] Aziz, A.: Integral mean estimates for polynomials with restricted zeros.J. Approx. Theory 55 (1988), 232–239. MR 0965219, 10.1016/0021-9045(88)90089-5 |
| Reference:
|
[4] Boas, R.P., Rahman, Q.I.: $L^p$ inequalities for polynomials and entire function.Arch. Rational Mech. Anal. 11 (1962), 34–39. MR 0158994, 10.1007/BF00253927 |
| Reference:
|
[5] Pólya, G., Szegö, G.: Aufgaben and Lehrsätze aus der Analysis.Springer-Verlag, Berlin, 1925. MR 0015435 |
| Reference:
|
[6] Rahman, Q.I., Schmeisser, G.: Analytic theory of polynomials.Oxford University Press, 1922. MR 1954841 |
| Reference:
|
[7] Rahman, Q.I., Schmeisser, G.: $L^{p}$ inequalities for polynomials.J. Approx. Theory 53 (1988), 26–32. MR 0937140, 10.1016/0021-9045(88)90073-1 |
| Reference:
|
[8] Royden, H.L.: Real Analysis.Macmillan Pub. Co., Inc., New York, 1968. MR 0151555 |
| Reference:
|
[9] Turán, P.: Über die Ableitung von Polynomen.Compositio Math. 7 (1939), 89–95. MR 0000228 |
| . |
