\documentclass{amsart} % options include 12pt or 11pt or 10pt % classes include article, report, book, letter, thesis \usepackage{setspace} \usepackage{a4} \usepackage{amsthm} \usepackage{latexsym} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{textcomp} \usepackage{cite} \usepackage{enumerate} \usepackage{amssymb} \usepackage{hyperref} \usepackage{amsmath} \usepackage{tikz} \usepackage[mathscr]{euscript} \usepackage{mathtools} \newtheorem{theorem}{Theorem}[section] \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem] {Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{lemma} [theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{observation}[theorem]{Observation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \setlength{\parindent}{0pt} \setlength{\evensidemargin}{0.3cm} \setlength{\oddsidemargin}{0.3cm} \setlength{\topmargin}{-1cm} \textwidth 16cm \textheight 23cm \onehalfspacing \title{This is the title} \newcommand{\Cross}{\mathbin{\tikz [x=1.4eh,y=1.4eh,line width=.2ex] \draw (0,0) -- (1,1) (0,1) -- (1,0);}}% \raggedbottom \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \usepackage{fancyhdr} %\pagestyle{fancy} \pagestyle{fancy} %\thispagestyle{empty} \fancyhead[LO]{\textbf{Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds for Roots of Polynomials}} \fancyhead[RE]{\textbf{K. Mahesh Krishna}} \begin{document} \hrule\hrule\hrule\hrule\hrule \vspace{0.3cm} \begin{center} {\bf\large{{Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds for Roots of Polynomials}}}\\ \vspace{0.3cm} \hrule\hrule\hrule\hrule\hrule \vspace{0.3cm} \textbf{K. Mahesh Krishna}\\ School of Mathematics and Natural Sciences\\ Chanakya University Global Campus\\ NH-648, Haraluru Village\\ Devanahalli Taluk, Bengaluru North District\\ Karnataka State, 562 110, India\\ Email: kmaheshak@gmail.com\\ Date: \today \end{center} \hrule\hrule \vspace{0.5cm} %-------------------------------------- \textbf{Abstract}: Two centuries ago, Lagrange derived an upper bound for every root of a complex polynomial using the maximum of powers of coefficients. In 1916, Fujiwara independently found the Lagrange bound. In 1914, Carmichael and Mason derived an upper bound using the sum of squares of the absolute values of the coefficients of a polynomial. This bound was extended by Kuniyeda in 1916. We derive noncommutative versions of the Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda bounds. \textbf{Keywords}: Lagrange-Fujiwara bound, Carmichael-Mason bound, Kuniyeda bound, Banach algebra. \textbf{Mathematics Subject Classification (2020)}: 30C15, 46H05, 46L05.\\ \hrule %\tableofcontents \hrule \section{Introduction} More than 200 years ago, Lagrange derived an upper bound for the zeros of complex polynomials \cite{OSTROWSKI, YAP}. This bound was later found independently by Fujiwara \cite{FUJIWARA2} in 1916. \begin{theorem} \label{LUB}\cite{OSTROWSKI, YAP, FUJIWARA2, GORA} (\textbf{Lagrange-Fujiwara Upper Bound}) Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in \mathbb{C}[z]. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|< 2 \max_{0\leq j \leq n-1}|a_j|^\frac{1}{n-j}. \end{align*} \end{theorem} \begin{corollary}\label{LUPG} \cite{OSTROWSKI, YAP, FUJIWARA2, GORA} (\textbf{Lagrange-Fujiwara Upper Bound}) Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathbb{C}[z], \quad a_n \neq 0. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|< 2 \max_{0\leq j \leq n-1}\left|\frac{a_j}{a_n}\right|^\frac{1}{n-j}. \end{align*} \end{corollary} Corollary \ref{LUPG} gives the following result. \begin{theorem}\cite{OSTROWSKI, YAP, FUJIWARA2, GORA} (\textbf{Lagrange-Fujiwara Lower Bound}) \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathbb{C}[z], \quad a_0\neq 0, a_n \neq 0. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|>\frac{1}{2\max_{1\leq j \leq n}\left|\frac{a_{j}}{a_0}\right|^\frac{1}{j}}. \end{align*} \end{theorem} In 1914, Carmichael and Mason derived the following beautiful upper bound (using Hadamard theorem for determinants) for every root of a complex polynomial using the sum of the squares of the absolute values of the coefficients \cite{CARMICHAELMASON}. \begin{theorem} \cite{CARMICHAELMASON, RAHMANSCHMEISSER, MARDEN, FUJIWARA} \label{CMB} (\textbf{Carmichael-Mason Upper Bound}) Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in \mathbb{C}[z]. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|< \left(1+ \sum_{j=0}^{n-1}|a_j|^2\right)^\frac{1}{2}. \end{align*} \end{theorem} In 1916, Kuniyeda improved Theorem \ref{CMB} using the Holder inequality \cite{KUNIYEDA}. \begin{theorem} \cite{KUNIYEDA, RAHMANSCHMEISSER, MARDEN, BORWEINERDELYI, TOYA, MIGNOTTESTEFANESCU, ABUOMAR} \label{KB} (\textbf{Kuniyeda Upper Bound}) Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in \mathbb{C}[z]. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|< \left(1+\left(\sum_{j=0}^{n-1}|a_j|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}, \quad \forall s \in (1, \infty). \end{align*} \end{theorem} \begin{corollary} \cite{ABUOMAR, KUNIYEDA, RAHMANSCHMEISSER, MARDEN, BORWEINERDELYI, TOYA, MIGNOTTESTEFANESCU} \label{KUB} (\textbf{Kuniyeda Upper Bound}) Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathbb{C}[z], \quad a_n \neq 0. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|< \left(1+\left(\sum_{j=0}^{n-1}\left|\frac{a_j}{a_n}\right|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}=\left(1+\left(\frac{1}{|a_n|^s}\sum_{j=0}^{n-1}|a_j|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}, \quad \forall s \in (1, \infty). \end{align*} \end{corollary} Corollary \ref{KUB} gives the following result. \begin{theorem}\label{CLB} \cite{ABUOMAR, KUNIYEDA, RAHMANSCHMEISSER, MARDEN, BORWEINERDELYI, TOYA, MIGNOTTESTEFANESCU} (\textbf{Kuniyeda Lower Bound}) Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathbb{C}[z], \quad a_0\neq 0, a_n \neq 0. \end{align*} If $\lambda \in \mathbb{C}$ satisfies $p(\lambda)=0$, then \begin{align*} |\lambda|&>\frac{1}{\left(1+\left(\frac{1}{|a_0|^s}\sum_{j=1}^{n}|a_j|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}}, \quad \forall s \in (1, \infty). \end{align*} \end{theorem} We naturally ask for versions of Theorems \ref{LUB} and \ref{KB} for polynomials over Banach algebras (particularly, over C*-algebras). In this article, we derive them. \section{Noncommutative Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda Bounds} Let $\mathcal{A}$ be a unital Banach algebra and $\mathcal{A}[z]$ be the set of all polynomials over $\mathcal{A}$. We derive a noncommutative version of Theorem \ref{LUB} as follows. \begin{theorem} (\textbf{Noncommutative Lagrange-Fujiwara Upper Bound})\label{NLFB} Let $\mathcal{A}$ be a unital Banach algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in \mathcal{A}[z]. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$ and \begin{align}\label{EL} \|u^n\|=\|u\|^n. \end{align} Then \begin{align*} \|u\|< 2 \max_{0\leq j \leq n-1}\|a_j\|^\frac{1}{n-j}. \end{align*} \end{theorem} \begin{proof} Let $u \in \mathcal{A}$ satisfies $p(u)=0$ and $\|u^n\|=\|u\|^n$. Define \begin{align*} M \coloneqq \max_{0\leq j \leq n-1}\|a_j\|^\frac{1}{n-j}. \end{align*} Then we have \begin{align}\label{PI} M^{n-j}\geq \|a_j\|, \quad \forall 0\leq j \leq n-1. \end{align} If $\|u\|\leq M$, then the conclusion holds. Hence we assume that $\|u\|>M$. Since $p(u)=0$, we have \begin{align*} u^n=-(a_0+a_1u+\cdots+a_{n-1}u^{n-1}). \end{align*} By taking the norm and using Equation (\ref{EL}), we get \begin{align*} \|u\|^n&=\|u^n\|=\|-(a_0+a_1u+\cdots+a_{n-1}u^{n-1})\|\\ &=\|a_0+a_1u+\cdots+a_{n-1}u^{n-1}\|\\ &\leq \|a_0\|+\|a_1u\|+\cdots+\|a_{n-1}u^{n-1}\|\\ &\leq \|a_0\|+\|a_1\|\|u\|+\cdots+\|a_{n-1}\|\|u^{n-1}\|\\ &\leq \|a_0\|+\|a_1\|\|u\|+\cdots+\|a_{n-1}\|\|u\|^{n-1}. \end{align*} Dividing by $\|u\|^n$ and by using Equation (\ref{PI}) we get \begin{align*} 1&\leq \frac{\|a_0\|}{\|u\|^n}+\frac{\|a_1\|}{\|u\|^{n-1}}+\cdots+ \frac{\|a_{n-2}\|}{\|u\|^{2}}+\frac{\|a_{n-1}\|}{\|u\|^{1}}\\ &\leq \frac{M^n}{\|u\|^n}+\frac{M^{n-1}}{\|u\|^{n-1}}+\cdots+ \frac{M^2}{\|u\|^{2}}+\frac{M}{\|u\|^{1}}\\ &=\frac{M}{\|u\|}\left(\left(\frac{M}{\|u\|}\right)^{n-1}+\left(\frac{M}{\|u\|}\right)^{n-2}+\cdots+\frac{M}{\|u\|}+1 \right)\\ &=\frac{M}{\|u\|}\left(\frac{1-\left(\frac{M}{\|u\|}\right)^{n}}{1-\frac{M}{\|u\|}}\right)\\ &<\frac{M}{\|u\|}\left(\frac{1}{1-\frac{M}{\|u\|}}\right)\\ &=\frac{M}{\|u\|-M}. \end{align*} Hence \begin{align*} 1<\frac{M}{\|u\|-M}. \end{align*} Rearranging completes the proof. \end{proof} Note that we used the condition (\ref{EL}) in the proof of Theorem \ref{NLFB}. We are unable to derive the result without this assumption. \begin{corollary}\label{CLUB} (\textbf{Noncommutative Lagrange-Fujiwara Upper Bound}) Let $\mathcal{A}$ be a unital Banach algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathcal{A}[z], \quad a_n \text{ is invertible}. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$ and \begin{align*} \|u^n\|=\|u\|^n. \end{align*} Then \begin{align*} \|u\|< 2 \max_{0\leq j \leq n-1}\|a_n^{-1}a_j\|^\frac{1}{n-j}. \end{align*} \end{corollary} \begin{corollary} Let $\mathcal{A}$ be a unital C*-algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathcal{A}[z], \quad a_n \text{ is invertible}. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$ and $u$ is normal or self-adjoint. Then \begin{align*} \|u\|< 2 \max_{0\leq j \leq n-1}\|a_n^{-1}a_j\|^\frac{1}{n-j}. \end{align*} \end{corollary} \begin{corollary} (\textbf{Noncommutative Lagrange-Fujiwara Lower Bound}) Let $\mathcal{A}$ be a unital Banach algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathcal{A}[z], \quad a_0\text{ and } a_n \text{ are invertible}. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$, $u$ is invertible and \begin{align*} \|(u^{-1})^n\|=\|u^{-1}\|^n. \end{align*} Then \begin{align*} \frac{1}{\|u^{-1}\|}>\frac{1}{2 \max_{1\leq j \leq n}\|a_0^{-1}a_{j}\|^\frac{1}{j}}. \end{align*} \end{corollary} \begin{proof} Since $u$ is invertible, we have \begin{align*} 0=a_n+a_{n-1}u^{-1}+\cdots+a_1u^{-(n-1)}+a_0u^{-n}. \end{align*} Define $v\coloneqq u^{-1}$. Then \begin{align*} 0=a_n+a_{n-1}v+\cdots+a_1v^{n-1}+a_0v^n. \end{align*} Result follows by applying Corollary \ref{CLUB} to $v$. \end{proof} We derive a noncommutative version of Theorem \ref{KB} as follows. \begin{theorem} \label{NCB} (\textbf{Noncommutative Kuniyeda Upper Bound}) Let $\mathcal{A}$ be a unital Banach algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+z^n\in \mathcal{A}[z]. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$ and \begin{align}\label{E1} \|u^n\|=\|u\|^n. \end{align} Then \begin{align*} \|u\|< \left(1+\left(\sum_{j=0}^{n-1}\|a_j\|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}, \quad \forall s \in (1, \infty). \end{align*} \end{theorem} \begin{proof} Let $s \in (1, \infty)$. Define \begin{align*} M\coloneqq \left(\sum_{j=0}^{n-1}\|a_j\|^s\right)^\frac{1}{s-1}. \end{align*} Let $u \in \mathcal{A}$ satisfies $p(u)=0$ and $\|u^n\|=\|u\|^n$. We claim that \begin{align*} \|u\|<\left(1+M\right)^\frac{s-1}{s}. \end{align*} Let us suppose that the claim fails. Then \begin{align*} \|u\|\geq \left(1+M\right)^\frac{s-1}{s}>1. \end{align*} Since $p(u)=0$, we have \begin{align*} u^n=-(a_0+a_1u+\cdots+a_{n-1}u^{n-1}). \end{align*} By taking the norm, using the Holder inequality and using Equation (\ref{E1}), we get \begin{align*} \|u\|^n&=\|u^n\|=\|-(a_0+a_1u+\cdots+a_{n-1}u^{n-1})\|\\ &=\|a_0+a_1u+\cdots+a_{n-1}u^{n-1}\|\\ &\leq \|a_0\|+\|a_1u\|+\cdots+\|a_{n-1}u^{n-1}\|\\ &\leq \|a_0\|+\|a_1\|\|u\|+\cdots+\|a_{n-1}\|\|u^{n-1}\|\\ &\leq \|a_0\|+\|a_1\|\|u\|+\cdots+\|a_{n-1}\|\|u\|^{n-1}\\ &=\sum_{j=0}^{n-1}\|a_j\|\|u\|^{j}\\ &\leq \left(\sum_{j=0}^{n-1}\|a_j\|^s\right)^\frac{1}{s}\left(\sum_{j=0}^{n-1}\left(\|u\|^j\right)^\frac{s}{s-1}\right)^\frac{s-1}{s}\\ &=M^\frac{s-1}{s}\left(\sum_{j=0}^{n-1}\left(\|u\|^\frac{s}{s-1}\right)^j\right)^\frac{s-1}{s}\\ &=M^\frac{s-1}{s}\left(\frac{\left(\|u\|^\frac{s}{s-1}\right)^n-1}{\|u\|^\frac{s}{s-1}-1}\right)^\frac{s-1}{s}\\ &=M^\frac{s-1}{s}\left(\frac{\|u\|^\frac{ns}{s-1}-1}{\|u\|^\frac{s}{s-1}-1}\right)^\frac{s-1}{s}. \end{align*} Hence \begin{align*} \|u\|^\frac{ns}{s-1}\leq M\frac{\|u\|^\frac{ns}{s-1}-1}{\|u\|^\frac{s}{s-1}-1}. \end{align*} Rearranging, \begin{align*} \|u\|^\frac{s}{s-1}-1\leq M\frac{\|u\|^\frac{ns}{s-1}-1}{\|u\|^\frac{ns}{s-1}}=M \left(1-\frac{1}{\|u\|^\frac{ns}{s-1}}\right)\frac{1}{\left(1+\left(\sum_{j=1}^{n}\|a_0^{-1}a_j\|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}}\geq \frac{1}{\left(1+\left(\|a_0^{-1}\|^s\sum_{j=1}^{n}\|a_j\|^s\right)^\frac{1}{s-1}\right)^\frac{s-1}{s}}. \end{align*} \end{corollary} \begin{corollary} (\textbf{Noncommutative Carmichael-Mason Upper Bound}) Let $\mathcal{A}$ be a unital Banach algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathcal{A}[z], \quad a_n \text{ is invertible}. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$ and \begin{align*} \|u^n\|=\|u\|^n. \end{align*} Then \begin{align*} \|u\|< \left(1+\sum_{j=0}^{n-1}\|a_n^{-1}a_j\|^2\right)^\frac{1}{2}\leq\left(1+\|a_n^{-1}\|^2\sum_{j=0}^{n-1}\|a_j\|^2\right)^\frac{1}{2}. \end{align*} \end{corollary} \begin{corollary} (\textbf{Noncommutative Carmichael-Mason Lower Bound}) Let $\mathcal{A}$ be a unital Banach algebra. Let \begin{align*} p(z)=a_0+a_1z+\cdots+a_{n-1}z^{n-1}+a_nz^n\in \mathcal{A}[z], \quad a_0\text{ and } a_n \text{ are invertible}. \end{align*} Let $u \in \mathcal{A}$ be such that $p(u)=0$, $u$ is invertible and \begin{align*} \|(u^{-1})^n\|=\|u^{-1}\|^n. \end{align*} Then \begin{align*} \frac{1}{\|u^{-1}\|}> \frac{1}{\left(1+\sum_{j=1}^{n}\|a_0^{-1}a_j\|^2\right)^\frac{1}{2}}\geq \frac{1}{\left(1+\|a_0^{-1}\|^2\sum_{j=1}^{n}\|a_j\|^2\right)^\frac{1}{2}}. \end{align*} \end{corollary} \section{Conclusions} \begin{enumerate} \item Over two centuries ago, Lagrange established an upper bound for the zeros of complex polynomials \cite{OSTROWSKI, YAP}. In 1916, Fujiwara found this bound independently \cite{FUJIWARA2}. \item In 1914, Carmichael and Mason derived an upper bound for the zeros of complex polynomials \cite{CARMICHAELMASON}. \item In 1916, Kuniyeda improved Carmichael-Mason bound \cite{KUNIYEDA}. \item In this article, we derived noncommutative versions of Lagrange-Fujiwara and Kuniyeda bounds. \end{enumerate} \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{ABUOMAR} Amer Abu-Omar. \newblock Notes on some bounds for the zeros of polynomials. \newblock {\em Math. Inequal. Appl.}, 21(2):481--487, 2018. \bibitem{BORWEINERDELYI} Peter Borwein and Tam{\'a}s Erd{\'e}lyi. \newblock {\em Polynomials and polynomial inequalities}. \newblock New York, NY: Springer-Verlag, 1995. \bibitem{CARMICHAELMASON} R.~D. Carmichael and T.~E. Mason. \newblock Note on the roots of algebraic equations. \newblock {\em Bull. Am. Math. 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