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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].
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].
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.