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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].
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.
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.
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].