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

Deepak Kumar, Dinesh Tripathi, Sunil Hans

Abstract

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.

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

Abstract

Let be a polynomial of degree . In this paper, we consider the modified Smirnov operator, which carries a polynomial into where is an arbitrary number in . We estimate minimum and maximum moduli of modified Smirnov operator of 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.

Paper Structure

This paper contains 4 sections, 16 theorems, 57 equations.

Key Result

Theorem 1

Let $F(z)$ be a polynomial of degree $n$, having all its zeros in $\overline{\mathbb{D}}$ and $P(z)$ be a polynomial of degree not exceeding that of $F(z)$. If $|P(z)|\leq |F(z)|$ on $B(\mathbb{D})$, then The equality holds only if $P(z)=e^{i\gamma}F(z), \gamma \in \mathbb{R}$.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Corollary 8
  • Corollary 9
  • Corollary 10
  • ...and 11 more