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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 Lq INEQUALITIES FOR POLYNOMIAL
Barchand Chanam,N. Reingachan,Khangembam Babina Devi,Maisnam Triveni Devi,Kshetrimayum Krishnadas 경남대학교 수학교육과 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 수학교육과 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].
BERNSTIEN AND TURÁN TYPE INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL
N. Reingachan,Barchand Chanam 경남대학교 수학교육과 2023 Nonlinear Functional Analysis and Applications Vol.28 No.1
The goal of this paper is to extend some inequalities of Bernstein as well as Tur\'{a}n type to polar derivative of a polynomial.
N. Reingachan,Robinson Soraisam,Barchand Chanam 경남대학교 수학교육과 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.
SOME INEQUALITIES ON POLAR DERIVATIVE OF A POLYNOMIAL
N. Reingachan,Robinson Soraisam,Barchand Chanam 경남대학교 수학교육과 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.
L^r LINEQUALITIES OF GENERALIZED TURÁN-TYPE INEQUALITIES OF POLYNOMIALS
Thangjam Birkramjit Singh,Kshetrimayum Krishnadas,Barchand Chanam 경남대학교 수학교육과 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].
TURÁN-TYPE $L^r$-INEQUALITIES FOR POLAR DERIVATIVE OF A POLYNOMIAL
Robinson Soraisam,Mayanglambam Singhajit Singh,Barchand Chanam 경남대학교 수학교육과 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.
IMPROVED BOUNDS OF POLYNOMIAL INEQUALITIES WITH RESTRICTED ZERO
Robinson Soraisam,Nirmal Kumar Singha,Barchand Chanam 경남대학교 수학교육과 2023 Nonlinear Functional Analysis and Applications Vol.28 No.2
In this paper, we shall first improve as well as generalize the above inequality. Further, we also improve the bounds of two known inequalities obtained by Govil et al. [8].
IMPROVEMENT AND GENERALIZATION OF A THEOREM OF T. J. RIVLIN
Mahajan Pritika,Khangembam Babina Devi,N. Reingachan,Barchand Chanam 경남대학교 수학교육과 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.