arXiv subjects

Danish Rashid Bhat

Publications and source records attributed to Danish Rashid Bhat.

4 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

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

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

A refinement of Cauchys theorem on the zeros of quaternion polynomial

In this paper, we shall present an interesting and significant refinement of a classical result of Cauchy about the moduli of the zeros of a quaternionic polynomial. As an application of this result we shall obtain zero-free regions of polynomials having quaternionic coefficients.

math.CV↗