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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.
IMPROVED VERSION ON SOME INEQUALITIES OF A POLYNOMIAL
Rashmi Rekha Sahoo,N. Reingachan,Robinson Soraisam,Khangembam Babina Devi,Barchand Chanam 경남대학교 수학교육과 2023 Nonlinear Functional Analysis and Applications Vol.28 No.4
Let $P(z)$ be a polynomial of degree $n$ and $P(z)\neq0$ in $|z|<1$. Then for every real $\alpha$ and $R>1$,Aziz \cite{Aziz1} proved that$$ \max_{|z|=1}|P(Rz)-P(z)|\leq \frac{R^{n}-1}{2}\left(M_{\alpha}^{2}+M_{\alpha+\pi}^{2}\right)^{\frac{1}{2}},$$where\begin{equation*}M_{\alpha}=\max_{1\leq k\leq n}|P(e^{i(\alpha+2k\pi) n})|. \end{equation*}\par In this paper, we establish some improvements and generalizations of the above inequality concerning the polynomials and their ordinary derivatives.
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.
HIGHER DERIVATIVE VERSIONS ON THEOREMS OF S. BERNSTEIN
Thangjam Birkramjit Singh,Khangembam Babina Devi,N. Reingachan,Robinson Soraisam,Barchand Chanam 경남대학교 수학교육과 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.
L^r INEQUALITIES FOR POLYNOMIALS
Reingachan Ngamchui,M. S. Singh,N. K. Singha,K. B. Devi,B. Chanam 경남대학교 수학교육과 2024 Nonlinear Functional Analysis and Applications Vol.29 No.2
In this paper, we not only obtain the $L^{r}$ version of the polar derivative of the above inequality for $r>0$, but also obtain an improved $L^{r}$ extension in polar derivative.
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.