Inequalities for B-operator
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Let $\mathcal{P}_n$ denote the space of all complex polynomials $P(z)=\sum\limits_{j=0}^{n}a_j z^j$ of degree $n$. Let $P\in\mathcal{P}_n$, for any complex number $\alpha$, $D_\alpha P(z)=nP(z)+(\alpha -z)P'(z)$, denote the polar derivative of the polynomial $P(z)$ with respect to $\alpha$ and $B_n$ denote a family of operators that maps $\mathcal{P}_n$ into itself. In this paper, we combine the operators $B$ and $D_\alpha$ and establish certain operator preserving inequalities concerning polynomials, from which a variety of interesting results can be obtained as special cases.
Abstract
Let $\mathcal{P}_n$ denote the space of all complex polynomials $P(z)=\sum\limits_{j=0}^{n}a_j z^j$ of degree $n$. Let $P\in\mathcal{P}_n$, for any complex number $\alpha$, $D_\alpha P(z)=nP(z)+(\alpha -z)P'(z)$, denote the polar derivative of the polynomial $P(z)$ with respect to $\alpha$ and $B_n$ denote a family of operators that maps $\mathcal{P}_n$ into itself. In this paper, we combine the operators $B$ and $D_\alpha$ and establish certain operator preserving inequalities concerning polynomials, from which a variety of interesting results can be obtained as special cases.
참고문헌 (19)
- A. Aziz, "Inequalities for polar derivative of a polynomial", J. Approx. Theory, 55, pp.183~193 (1988)
- A. Aziz, "Inequalities for a polynomial and its derivatives", J. Approx. Theory, 54, pp.306~311 (1998)
- A. Aziz, "Shah, Inequalities for the polar derivative of a polynomial", Indian J. Pure Appl. Math, 29, pp.163~173 (1998)
- S. N. Bernstein, "Sur Lordre de la meilleur approximation des fonctions continues par des poly- ˇnomes de degr´e donn´e", M´emoire de ˇlAcad´emie Royal de Belgique, 4(2), pp.1~103 (1912)
- S. Bernstein, "Sur la limitation des derivees des polnomes", C. R. Acad. Sci. Paris, 190, pp.338~341 (1930)
- M. Bidkham, "An operator preserving inequalities between polar derivatives of a polynomial", J. Interdiscip. Math, 14, pp.591~601 (2011)
- K. K. Dewan, "Generalization of certain well-known polynomial inequalities", J. Math. Anal. Appl, 363, pp.38~41 (2010)
- M. H. Gulzar, "Inequalities for the polar derivative of a polynomial", J. Anal, 28, pp.923~929 (2020)
- M. H. Gulzar, "Some inequalities for the polar derivative of a polynomial", Kragujevac J. Math, 47, pp.567~576 (2023)
- E. Laguerre, "Oeuvres, Vol. I" (1898), Gauthier-Villars
- P. D. Lax, "Proof of a Conjecture of P. Erd¨os on the derivative of a polynomial", American Mathematical Society, 50(8), pp.509~513 (1994)
- A. Liman, "Inequalities for the polar derivative of a polynomial", Complex Anal. Oper. Theory, 6, pp.1199~1209 (2012)
- M. A. Malik, "Inequalities concerning the derivative of polynomials", Rend. Circ. Math. Palermo, 34(2), pp.422~426 (1985)
- M. Marden, "Geometry of polynomials" (1966), Amer.Math.Soc
- Q. I. Rahman, "Functions of exponential type", Trans. Amer. Math. Soc, 135, pp.295~309 (1969)
- Q. I. Rahman, "Analytic Theory of Polynomials" (2002), Oxford University Press
- W. M. Shah, "A generalization of a theorem of P. Turan", J. Ramanujan Math. Soc, 1, pp.29~35 (1996)
- W. M. Shah, "An operator preserving inequalities between polynomials", J. Inequalities Pure Applied Math, 9, pp.1~12 (2008)
- Xin Li, "A comparison inequality for rational functions", Proc. Am. Math. Soc, 139, pp.1659~1665 (2011)