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Suhail Gulzar

Publications and source records attributed to Suhail Gulzar.

12 recordsLinked to original sources
arXiv2024

On the zeros of certain composite polynomials and an operator preserving inequalities

If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}\lambda_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-\sigma|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm} that all the zeros of the polynomial $h(z)= \sum_{k=0}^{n}\lambda_k f^{(k)}(z) \frac{(\sigma z)^k}{k!}$ lie in the circle $|z| \leq r ~ \max(1,s)$. In this paper, we relax the condition that $f(z)$ and $g(z)$ are of the same degree and instead assume that $f(z)$ and $g(z)$ are polynomials of arbitrary degree $n$ and $m$ respectively, $m\leq n,$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserve Bernstein type polynomial inequalities.

math.CV↗
arXiv2024

On Visser's inequality concerning coefficient estimates for a polynomial

If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left\|a_nz+\frac{a_s}{\binom{n}{s}}\right\|_{p}\leq \frac{\left\|z+\delta_{0s}\right\|_p}{\left\|1+z\right\|_p}\left\|P\right\|_{p}, \end{align*} where $\delta_{0s}$ is the Kronecker delta. In this paper, we consider the class of polynomials having no zero in $|z|<\rho,$ $\rho\geq 1$ and obtain some generalizations of above inequality.

math.CV↗
arXiv2019

A note on a recent attempt to prove Sendov's conjecture

Recently GM Sofi & SA Shabir [arXive: 1903.01850v2 [math.GM] 6 Mar 2019] made an attempt to prove the Sendov's conjecture. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

math.CV↗
arXiv2015

Some extensions of Eneström-Kakeya Theorem

In this paper we obtain some refinements of a well-known result of Eneströ-Kakeya concerning the bounds for the moduli of the zeros of polynomials with complex coefficients which improve upon some results due to Aziz and Mohammad, Govil and Rahman and others.

math.CV↗
arXiv2014

On an inequality concerning the polar derivative of a polynomial with restricted zeros

Let $D_αP(z)=nP(z)+(α-z)P^{\prime}(z)$ denote the polar derivative of a polynomial $P(z)$ of degree $n$ with respect to a point $α\in\mathbb{C}.$ In this paper, we present a correct proof, independent of Laguerre's theorem, of an inequality concerning the polar derivative of a polynomial with restricted zeros recently formulated by K. K. Dewan, Naresh Singh, Abdullah Mir, [Extensions of some polynomial inequalities to the polar derivative, \emph{J. Math. Anal. Appl.,} \textbf{352} (2009) 807-815].

math.CV↗
arXiv2014

On the polar derivative of a polynomial

Let $P(z)$ be a polynomial of degree $n$ having no zero in $|z|<k$ where $k\geq 1,$ then for every real or complex number $α$ with $|α|\geq 1$ it is known \begin{equation*} \underset{|z|=1}{\max}|D_αP(z)|\leq n\left(\dfrac{|α|+k}{1+k}\right)\underset{|z|=1}{\max}|P(z)|, \end{equation*} where $D_αP(z)=nP(z)+(α-z)P^{\prime}(z)$ denote the polar derivative of the polynomial $P(z)$ of degree $n$ with respect to a point $α\in\mathbb{C}.$ In this paper, by a simple method, a refinement of above inequality and other related results are obtained.

math.CV↗
arXiv2013

On annulus containing all the zeros of a polynomial

In this paper, we obtain an annulus containing all the zeros of the polynomial involving binomial coefficients and generalized Fibonacci numbers. Our result generalize some of the recently obtained results in this direction.

math.CV↗
arXiv2013

Lp mean estimates for an operator preserving inequalities between polynomials

If $P(z)$ be a polynomial of degree at most $n$ which does not vanish in $|z| < 1$, it was recently formulated by Shah and Liman \cite[\textit{Integral estimates for the family of $B$-operators, Operators and Matrices,} \textbf{5}(2011), 79 - 87]{wl} that for every $R\geq 1$, $p\geq 1$, \[\left\|B[P\circσ](z)\right\|_p \leq\frac{R^{n}|Λ_n|+|λ_{0}|}{\left\|1+z\right\|_p}\left\|P(z)\right\|_p,\] where $B$ is a $ \mathcal{B}_{n}$-operator with parameters $λ_{0}, λ_{1}, λ_{2}$ in the sense of Rahman \cite{qir}, $σ(z)=Rz$ and $Λ_n=λ_{0}+λ_{1}\frac{n^{2}}{2} +λ_{2}\frac{n^{3}(n-1)}{8}$. Unfortunately the proof of this result is not correct. In this paper, we present a more general sharp $L_p$-inequalities for $\mathcal{B}_{n}$-operators which not only provide a correct proof of the above inequality as a special case but also extend them for $ 0 \leq p <1$ as well.

math.CV↗
arXiv2013

On an operator preserving inequalities between polynomials

Let $\mathscr{P}_n $ denote the space of all complex polynomials $P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$ and $\mathcal{B}_n$ a family of operators that maps $\mathscr{P}_n$ into itself. In this paper, we consider a problem of investigating the dependence of $$|B[P\circσ](z)-αB[P\circρ](z)+β\{(\frac{R+k}{k+r})^{n}-|α|\}B[P\circρ](z)| $$ on the maximum and minimum modulus of $|P(z)|$ on $|z|=k$ for arbitrary real or complex numbers $α,β\in\mathbb{C}$ with $|α|\leq 1,|β|\leq 1,R>r\geq k,$ $σ(z)=Rz,$ $ρ(z)=rz$ and establish certain sharp operator preserving inequalities between polynomials, from which a variety of interesting results follow as special cases.

math.CV↗
arXiv2013

Inequalities for the polar derivative of a polynomial

Let $ P(z) $ be a polynomial of degree $ n $ and for any real or complex number $α,$ let $D_αP(z)=nP(z)+(α-z)P^{\prime}(z)$ denote the polar derivative with respect to $α.$ In this paper, we obtain generalizations of some inequalities for the polar derivative of a polynomial.

math.CV↗
arXiv2013

Integral mean estimates for the polar derivative of a polynomial

Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} that for every real or complex number $α$ with $|α|\geq k$ and each $r\geq 0$ $$ n(|α|-k)\left\{\int\limits_{0}^{2π}\left|P\left(e^{iθ}\right)\right|^r dθ\right\}^{\frac{1}{r}}\leq\left\{\int\limits_{0}^{2π}\left|1+ke^{iθ}\right|^r dθ\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_αP(z)|. $$ \indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials $P(z)=a_nz^n+\sum_{ν=μ}^{n}a_{n-ν}z^{n-ν},$ $1\leqμ\leq n,$ having all its zeros in $|z|\leq k$ where $k\leq 1$ and thereby obtain certain generalizations of above and many other known results.

math.CV↗