INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL
If \(w(\zeta)\) is a polynomial of degree $n$ with all its zeros in \(|\zeta|\leq \Delta, \Delta\geq 1\) and any real \(\gamma\geq 1\), Aziz proved the integral inequality [1] $$ \left\lbrace\int_{0}^{2\pi}\left|1+\Delta^ne^{i\theta}\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}\max_{|\zeta|=1...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2024-12-01
|
| Series: | Ural Mathematical Journal |
| Subjects: | |
| Online Access: | https://umjuran.ru/index.php/umj/article/view/711 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849397234084347904 |
|---|---|
| author | Mayanglambam Singhajit Singh Barchand Chanam |
| author_facet | Mayanglambam Singhajit Singh Barchand Chanam |
| author_sort | Mayanglambam Singhajit Singh |
| collection | DOAJ |
| description | If \(w(\zeta)\) is a polynomial of degree $n$ with all its zeros in \(|\zeta|\leq \Delta, \Delta\geq 1\) and any real \(\gamma\geq 1\), Aziz proved the integral inequality [1] $$
\left\lbrace\int_{0}^{2\pi}\left|1+\Delta^ne^{i\theta}\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}\max_{|\zeta|=1}|w^{\prime}(\zeta)|\geq n\left\lbrace\int_{0}^{2\pi}\left|w\left(e^{i\theta}\right)\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}.
$$
In this article, we establish a refined extension of the above integral inequality by using the polar derivative instead of the ordinary derivative consisting of the leading coefficient and the constant term of the polynomial. Besides, our result also yields other intriguing inequalities as special cases. |
| format | Article |
| id | doaj-art-bc47a3c37b474ff281cd0852e56d376a |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-bc47a3c37b474ff281cd0852e56d376a2025-08-20T03:39:05ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522024-12-0110210.15826/umj.2024.2.012226INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIALMayanglambam Singhajit Singh0Barchand Chanam1Department of Mathematics, National Institute of Technology Manipur, Imphal, 795004Department of Mathematics, National Institute of Technology Manipur, Imphal, 795004If \(w(\zeta)\) is a polynomial of degree $n$ with all its zeros in \(|\zeta|\leq \Delta, \Delta\geq 1\) and any real \(\gamma\geq 1\), Aziz proved the integral inequality [1] $$ \left\lbrace\int_{0}^{2\pi}\left|1+\Delta^ne^{i\theta}\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}\max_{|\zeta|=1}|w^{\prime}(\zeta)|\geq n\left\lbrace\int_{0}^{2\pi}\left|w\left(e^{i\theta}\right)\right|^{\gamma}d\theta\right\rbrace^{{1}/{\gamma}}. $$ In this article, we establish a refined extension of the above integral inequality by using the polar derivative instead of the ordinary derivative consisting of the leading coefficient and the constant term of the polynomial. Besides, our result also yields other intriguing inequalities as special cases.https://umjuran.ru/index.php/umj/article/view/711polar derivative, turán-type inequalities, integral inequalities |
| spellingShingle | Mayanglambam Singhajit Singh Barchand Chanam INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL Ural Mathematical Journal polar derivative, turán-type inequalities, integral inequalities |
| title | INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL |
| title_full | INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL |
| title_fullStr | INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL |
| title_full_unstemmed | INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL |
| title_short | INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL |
| title_sort | integral analogue of turan type inequalities concerning the polar derivative of a polynomial |
| topic | polar derivative, turán-type inequalities, integral inequalities |
| url | https://umjuran.ru/index.php/umj/article/view/711 |
| work_keys_str_mv | AT mayanglambamsinghajitsingh integralanalogueofturantypeinequalitiesconcerningthepolarderivativeofapolynomial AT barchandchanam integralanalogueofturantypeinequalitiesconcerningthepolarderivativeofapolynomial |