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SOME Lq INEQUALITIES FOR POLYNOMIAL
Barchand Chanam,N. Reingachan,Khangembam Babina Devi,Maisnam Triveni Devi,Kshetrimayum Krishnadas 경남대학교 기초과학연구소 2021 Nonlinear Functional Analysis and Applications Vol.26 No.2
Let $p(z)$be a polynomial of degree n. Then Bernstein's inequality [12,18] is$$ \max_{|z|=1}|p^{'}(z)|\leq n\max_{|z|=1}|(z)|.$$For $q>0$, we denote$$\|p\|_{q}=\left\{\frac{1}{2\pi}\int^{2\pi}_{0}|p(e^{i\theta})|^{q}d\theta\right\}^{\frac{1}{q}},$$and a well-known fact from analysis [17] gives$$\lim_{q\rightarrow \infty} \left\{\frac{1}{2\pi} \int^{2\pi}_{0}|p(e^{i\theta})|^{q} d\theta\right\}^{\frac{1}{q}} = \max_{|z|=1}|p(z)|. $$ Above Bernstein's inequality was extended by Zygmund [19] into $L^{q}$ norm by proving\begin{equation*}\|p^{'}\|_{q}\leq n\|p\|_{q}, \;\;q\geq 1. \end{equation*} Let $p(z)=a_{0}+\sum^{n}_{\nu=\mu}a_{\nu}z^{\nu}$, $1\leq\mu\leq n$, be a polynomial of degree n having no zero in $|z|<k, k\geq 1$. Then for $0< r\leq R\leq k$, Aziz and Zargar [4] proved$$\max_{|z|=R}|p^{'}(z)|\leq \frac{nR^{\mu-1}(R^{\mu}+k^{\mu})^{\frac{n}{\mu}-1}}{(r^{\mu}+k^{\mu})^{\frac{n}{\mu}}}\max_{|z|=r}|p(z)|. $$ In this paper, we obtain the $L^{q}$ version of the above inequality for $q>0$. Further, we extend a result of Aziz and Shah [3] into $L^{q}$ analogue for $q>0$. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.
ON AN INEQUALITY OF S. BERNSTEIN
Barchand Chanam,Khangembam Babina Devi,Kshetrimayum Krishnadas,Maisnam Triveni Devi,Reingachan Ngamchui,Thangjam Birkramjit Singh 경남대학교 기초과학연구소 2021 Nonlinear Functional Analysis and Applications Vol.26 No.2
If $p(z)=\sum\limits_{\nu=0}^na_{\nu}z^{\nu}$ is a polynomial of degree $n$ having all its zeros on $|z|=k$, $k\leq 1$, then Govil [3]proved that\begin{align*}\max\limits_{|z|=1}|p'(z)|\leq \dfrac{n}{k^n+k^{n-1}}\max\limits_{|z|=1}|p(z)|. \end{align*} In this paper, by involving certain coefficients of $p(z)$, we not only improve the above inequality but also improve a result provedby Dewan and Mir [2].
Remark on Some Recent Inequalities of a Polynomial and its Derivatives
Barchand Chanam,Khangembam Babina Devi,Thangjam Birkramjit Singh 경북대학교 자연과학대학 수학과 2022 Kyungpook mathematical journal Vol.62 No.3
We point out a flaw in a result proved by Singh and Shah [Kyungpook Math. J., 57(2017), 537-543] which was recently published in Kyungpook Mathematical Journal. Further, we point out an error in another result of the same paper which we correct and obtain integral extension of the corrected form.
SOME INEQUALITIES ON POLAR DERIVATIVE OF A POLYNOMIAL
N. Reingachan,Robinson Soraisam,Barchand Chanam 경남대학교 기초과학연구소 2022 Nonlinear Functional Analysis and Applications Vol.27 No.4
In this paper, we establish some extensions and refinements of the above inequality topolar derivative and some other well-known inequalities concerning the polynomials and theirordinary derivatives.
N. Reingachan,Robinson Soraisam,Barchand Chanam 경남대학교 기초과학연구소 2022 Nonlinear Functional Analysis and Applications Vol.27 No.4
In this paper, we present some fixed point theorems for rational type contractiveconditions in the setting of a complete metric space via a cyclic (α, β)-admissible mapping imbedded in simulation function. Our results extend and generalize some previous works from the existing literature. We also give some examples to illustrate the obtained results.
TURÁN-TYPE $L^r$-INEQUALITIES FOR POLAR DERIVATIVE OF A POLYNOMIAL
Robinson Soraisam,Mayanglambam Singhajit Singh,Barchand Chanam 경남대학교 기초과학연구소 2023 Nonlinear Functional Analysis and Applications Vol.28 No.3
In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. \cite{SLBEPD2021}into $L^{r}$-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.
IMPROVEMENT AND GENERALIZATION OF A THEOREM OF T. J. RIVLIN
Mahajan Pritika,Khangembam Babina Devi,N. Reingachan,Barchand Chanam 경남대학교 기초과학연구소 2022 Nonlinear Functional Analysis and Applications Vol.27 No.3
In this paper, we generalize as well as sharpen the above inequality. Also our results not only generalize, but also sharpen some known results proved recently.
L^r LINEQUALITIES OF GENERALIZED TURÁN-TYPE INEQUALITIES OF POLYNOMIALS
Thangjam Birkramjit Singh,Kshetrimayum Krishnadas,Barchand Chanam 경남대학교 기초과학연구소 2021 Nonlinear Functional Analysis and Applications Vol.26 No.4
If $p(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\leq 1$, then for $\rho R\geq k^2$ and $\rho\leq R$, Aziz and Zargar [4] proved that\max\limits_{|z|=1}|p'(z)|\geq n\dfrac{(R+k)^{n-1}}{(\rho+k)^n}\left\{\max\limits_{|z|=1}|p(z)|+\min\limits_{|z|=k}|p(z)|\right\}. We prove a generalized $L^r$ extension of the above result for a more general class of polynomials $p(z)=a_nz^n+\sum\limits_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu}$, $1\leq \mu\leq n$. We also obtain another $L^r$ analogue of a result for the above general class of polynomials proved by Chanam and Dewan [6].
HIGHER DERIVATIVE VERSIONS ON THEOREMS OF S. BERNSTEIN
Thangjam Birkramjit Singh,Khangembam Babina Devi,N. Reingachan,Robinson Soraisam,Barchand Chanam 경남대학교 기초과학연구소 2022 Nonlinear Functional Analysis and Applications Vol.27 No.2
In this paper, we first prove a result concerning the sth derivative where 1 ≤ s < n of the polynomial involving some of the co-efficients of the polynomial. Our result not only improves and generalizes the above inequality, but also gives a generalization to higher derivative of a result due to Dewan and Mir [2] in this direction. Further, a direct generalization of the above inequality for the sth derivative where 1 ≤ s < n is also proved.
IMPROVEMENT AND GENERALIZATION OF POLYNOMIAL INEQUALITY DUE TO RIVLIN
Nirmal Kumar Singha,Reingachan Ngamchui,Maisnam Triveni Devi,Barchand Chanam 경남대학교 기초과학연구소 2023 Nonlinear Functional Analysis and Applications Vol.28 No.3
Let $p(z)$ be a polynomial of degree $n$ having no zero in $\vert z\vert<1$. In this paper, by involving some coefficients of the polynomial, we prove an inequality that not only improves as well as generalizes the well-known result proved by Rivlin but also has some interesting consequences.