arXiv subjects
Publications and source records attributed to N. A. Rather.
In this paper, we obtain an annulus containing all the zeros of the polynomial involving binomial coefficients and generalized Fibonacci numbers. Our result generalize some of the recently obtained results in this direction.
If $P(z)$ be a polynomial of degree at most $n$ which does not vanish in $|z| < 1$, it was recently formulated by Shah and Liman \cite[\textit{Integral estimates for the family of $B$-operators, Operators and Matrices,} \textbf{5}(2011), 79 - 87]{wl} that for every $R\geq 1$, $p\geq 1$, \[\left\|B[P\circσ](z)\right\|_p \leq\frac{R^{n}|Λ_n|+|λ_{0}|}{\left\|1+z\right\|_p}\left\|P(z)\right\|_p,\] where $B$ is a $ \mathcal{B}_{n}$-operator with parameters $λ_{0}, λ_{1}, λ_{2}$ in the sense of Rahman \cite{qir}, $σ(z)=Rz$ and $Λ_n=λ_{0}+λ_{1}\frac{n^{2}}{2} +λ_{2}\frac{n^{3}(n-1)}{8}$. Unfortunately the proof of this result is not correct. In this paper, we present a more general sharp $L_p$-inequalities for $\mathcal{B}_{n}$-operators which not only provide a correct proof of the above inequality as a special case but also extend them for $ 0 \leq p <1$ as well.
Let $\mathscr{P}_n $ denote the space of all complex polynomials $P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$ and $\mathcal{B}_n$ a family of operators that maps $\mathscr{P}_n$ into itself. In this paper, we consider a problem of investigating the dependence of $$|B[P\circσ](z)-αB[P\circρ](z)+β\{(\frac{R+k}{k+r})^{n}-|α|\}B[P\circρ](z)| $$ on the maximum and minimum modulus of $|P(z)|$ on $|z|=k$ for arbitrary real or complex numbers $α,β\in\mathbb{C}$ with $|α|\leq 1,|β|\leq 1,R>r\geq k,$ $σ(z)=Rz,$ $ρ(z)=rz$ and establish certain sharp operator preserving inequalities between polynomials, from which a variety of interesting results follow as special cases.
Let $ P(z) $ be a polynomial of degree $ n $ and for any real or complex number $α,$ let $D_αP(z)=nP(z)+(α-z)P^{\prime}(z)$ denote the polar derivative with respect to $α.$ In this paper, we obtain generalizations of some inequalities for the polar derivative of a polynomial.
Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} that for every real or complex number $α$ with $|α|\geq k$ and each $r\geq 0$ $$ n(|α|-k)\left\{\int\limits_{0}^{2π}\left|P\left(e^{iθ}\right)\right|^r dθ\right\}^{\frac{1}{r}}\leq\left\{\int\limits_{0}^{2π}\left|1+ke^{iθ}\right|^r dθ\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_αP(z)|. $$ \indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials $P(z)=a_nz^n+\sum_{ν=μ}^{n}a_{n-ν}z^{n-ν},$ $1\leqμ\leq n,$ having all its zeros in $|z|\leq k$ where $k\leq 1$ and thereby obtain certain generalizations of above and many other known results.
In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.