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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.
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.