Searcharxiv⌕ Search

arXiv subjects

N. A. Rather

Publications and source records attributed to N. A. Rather.

At least 19 recordsLinked to original sources
arXiv2026

Inequalities For The Growth Of Rational Functions With Prescribed Poles

Let $\mathcal R_{n}$ be the set of all rational functions of the type $r(z) = f(z)/w(z)$, where $f(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-β_j)$, $|β_j|>1$ for $1\leq j\leq n$. In this work, we investigate the growth behavior of rational functions with prescribed poles by utilizing certain coefficients of the polynomial $f(z)$. The results obtained here not only refine and strengthen the findings of Rather et al. \cite{NS}, but also generalize recent growth estimates for polynomials due to Dhankhar and Kumar \cite{KD} to the broader setting of rational functions with fixed poles. Additionally, we establish corresponding results for such rational functions under suitable restrictions on their zeros.

math.CV↗
arXiv2026

Microscopic investigation of $γ~$ vibrational band structures in odd-mass nuclei

A systematic investigation of the high-spin band structures observed in $^{103,105,107,109}$Nb and $^{103,105,107,109}$Tc nuclides is performed using the triaxial projected shell model (TPSM) approach. For $^{103,105}$Nb isotopes, four bands have been populated with the lowest three bands corresponding to yrast, $γ$ and 2$γ$ bands. The nature of the fourth observed band has remained unresolved as it has been shown from the transition intensity ratios that this band cannot correspond to the expected 3$γ$ band. It is demonstrated in the present work that this fourth band is the second $γ$ band, resulting from the combination, $K=K_0-2$ with $K_0$ being the "$K$" value of the parent configuration. The excitation energy and other properties of this band structure are predicted for all the studied nuclides.

nucl-th↗
arXiv2026

Inequalities Concerning Rational Functions With Prescribed Poles

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results for rational functions with fixed poles and restricted zeros. The obtained results bring forth generalizations and refinements of some known inequalities for rational functions and in turn produce generalizations and refinements of some polynomial inequalities as well.

math.CV↗
arXiv2026

Microscopic investigation of magnetic and antimagnetic rotational motion in atomic nuclei

In the present work, we have generalized the projected shell model (PSM) approach to include the quasiparticle excitations from two major oscillator shells, and have also extended the basis space to five-quasiparticle configurations for odd-mass nuclei. The magnetic and antimagnetic rotational structures observed in odd-neutron Pd- and Cd-isotopes have been investigated as a first major application of the new development. It is shown that PSM approach provides a reasonable description of the observed properties of magnetic and antimagnetic rotational bands.

nucl-th↗
arXiv2026

A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial

Let \( P(z) \) be a polynomial of degree \( n \) and $α\in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_αP(z) \) and is defined by $D_αP(z): = nP(z) + (α-z)P'(z)$. The polar derivative \( D_αP(z) \) is a polynomial of degree at most \( n - 1 \) and it generalizes the ordinary derivative \( P'(z) \). In this paper, we establish some \( L_p \) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings.

math.CV↗
arXiv2025

On the location of zeros of a quaternion polynomial

In this paper, we are concerned with the problem of locating the zeros of polynomials of a quaternionic variable with quaternionic coefficients. We derive some new Cauchy bounds for the zeros of a polynomial by virtue of maximum modulus theorem. Our results will generalise some recently proved results about the distribution of zeros of a quaternionic polynomial.

math.CV↗
arXiv2024

Bounds for the Zeros of Quaternionic Polynomials and Regular Functions Using Matrix Techniques

We investigate the problem of determining the zeros of quaternionic polynomials using matrix method. In a recent paper, Dar et al. \cite{RD} proved that the zeros of a quaternionic polynomial and the left eigenvalues of the corresponding companion matrix are identical. Building on this, we employ various newly developed matrix techniques to establish several results concerning the location of the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients. These findings significantly enhance the understanding of quaternionic polynomials and their eigenvalues, offering a broader perspective on their mathematical properties.

math.CV↗
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 $g(z) = \sum_{k=0}^{n}λ_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-σ|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm} that all the zeros of the polynomial $h(z)= \sum_{k=0}^{n}λ_k f^{(k)}(z) \frac{(σz)^k}{k!}$ 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

Certain Bernstein-type $L_p$ inequalities for polynomials

Let $P(z)$ be a polynomial of degree $n,$ then it is known that for $α\in\mathbb{C}$ with $|α|\leq \frac{n}{2},$ \begin{align*} \underset{|z|=1}{\max}|\left|zP^{\prime}(z)-αP(z)\right|\leq \left|n-α\right|\underset{|z|=1}{\max}|P(z)|. \end{align*} This inequality includes Bernstein's inequality, concerning the estimate for $|P^\prime(z)|$ over $|z|\leq 1,$ as a special case. In this paper, we extend this inequality to $L_p$ norm which among other things shows that the condition on $α$ can be relaxed. We also prove similar inequalities for polynomials with restricted zeros.

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+δ_{0s}\right\|_p}{\left\|1+z\right\|_p}\left\|P\right\|_{p}, \end{align*} where $δ_{0s}$ is the Kronecker delta. In this paper, we consider the class of polynomials having no zero in $|z|<ρ,$ $ρ\geq 1$ and obtain some generalizations of above inequality.

math.CV↗
arXiv2024

Location of the zeros of quaternionic polynomials using matrix tools

Using a variety of matrix techniques, the problem of locating the left eigenvalues of the quaternion companion matrices are investigated in this paper. In a recent paper, Dar et al. [6], proved that the zeros of a quaternionic polynomial and the left eigenvalues of corresponding companion matrix are same. In view of this, we use various newly developed matrix techniques to prove various results concerning the location of the zeros of regular polynomials of a quaternionic variable with quaternionic coefficients, which include an extension of the result of A. L. Cauchy as well.

math.CV↗
arXiv2021

Inequalities for rational functions with prescribed poles

For rational functions, we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdös-Lax and Turán to rational functions R. In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdós-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we shall demonstrate how some recently obtained results due to S. L. Wali and W. M. Shah, [Some applications of Dubinin's lemma to rational functions with prescribed poles, J. Math.Anal.Appl.450 (2017) 769-779], could have been proved without invoking the results

math.CA↗
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↗