arXiv subjects

Sunil Hans

Publications and source records attributed to Sunil Hans.

3 recordsLinked to original sources

arXiv2025

Generalization of Some Well-Known Polynomial Inequalities for the Modified Smirnov Operator

Let $P(z)$ be a polynomial of degree $n$. In this paper, we consider the modified Smirnov operator, which carries a polynomial $P(z)$ into $\tilde{\mathbb{S}}_a[P](z)=(1+az)P'(z)-naP(z),$ where $a$ is an arbitrary number in $\overline{\mathbb{D}}$. We estimate minimum and maximum moduli of modified Smirnov operator of $P(z)$ on the unit circle with restricted zeros and thereby obtain a generalization of some results of Dewan and Hans \cite{dewan2013some}. This study includes compact generalization of some well-known polynomial inequalities.

math.CV↗

arXiv2025

Some Compact Generalization of Bernstein-Type Inequalities Preserved by Modified Smirnov Operator

Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of \[\bigg|P(Rz)-\alpha P(z)+\beta\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|\alpha|\biggr\}P(z)\bigg|, \ \text{for} \ z \in B(\mathbb{D}),\] on $\max_{z\in B(\mathbb{D})}|P(z)|$, for every real and complex number $\alpha, \beta$ satisfying $|\alpha| \leq 1$, $|\beta| \leq 1$, and $R \geq 1$. This paper presents a compact generalization of several well-known polynomial inequalities using modified Smirnov operator, demonstrating that the operator preserves inequalities between polynomials.

math.CV↗

arXiv2024

Generalizations of Some Inequalities of the Polar Derivatives of Polynomials

The main purpose of this paper is to present the generalization of the inequalities between the modulus of the polar derivative and the polynomial itself, depending on consideration of the zeros inside and outside of a closed disk and the extremal coefficient position of the underlying polynomial.

math.CV↗