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Rubia Akhter,M. H. Gulzar 강원경기수학회 2022 한국수학논문집 Vol.30 No.3
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.
INEQUALITIES CONCERNING POLYNOMIAL AND ITS DERIVATIVE
Zargar, B.A.,Gulzar, M.H.,Akhter, Tawheeda The Kangwon-Kyungki Mathematical Society 2021 한국수학논문집 Vol.29 No.3
In this paper, some sharp inequalities for ordinary derivative P'(z) and polar derivative D<sub>α</sub>P(z) = nP(z) + (α - z)P'(z) are obtained by including some of the coefficients and modulus of each individual zero of a polynomial P(z) of degree n not vanishing in the region |z| > k, k ≥ 1. Our results also improve the bounds of Turán's and Aziz's inequalities.
Rubia Akhter,B. A. Zargar,M. H. Gulzar 강원경기수학회 2023 한국수학논문집 Vol.31 No.2
In this paper, we study the growth of polynomials $P(z)$ of degree $n$ defined by $P(z)=z^s(a_0+\sum\limits_{j=t}^{n-s}a_j z^j),\quad t\geq 1,\quad 0\leq s\leq n-1$ which do not vanish in the disk $|z|\leq k, \quad k\geq 1$ except for the $s$-fold zeros at origin. Our result generalises and refines many results known in the literature.