Let $p(z)$ be a polynomial of degree $n$. Then the polar derivative with respect to real or complex number $\alpha$ is defined by $D_\alpha p(z)=n p(z)+(\alpha-z) p^{\prime}(z)$. Govil and Mctume [14] proved that if $n$ having all its zeros in $|z| \leq k, k \geq 1$, then for $|\alpha| 1+k+k^n$,$$\begin{aligned}\max _{|z|=1}\left|D_\alpha p(z)\right| & n\left(\frac{|\alpha|-k}{1+k^n}\right)...

Three unique diastereoisomers of 2,3,4,5-tetrafluorohexane have been prepared, compounds intermediate between hexane and perfluorohexane in their degree of fluorination, and they show very different conformational behaviour and physical properties.

In this paper, we consider the existence and asymptotic behaviours of solutions for following fractional equation$ \begin{align*} (-\Delta)^{s}u+\lambda u = \mu(I_{\alpha}\star|u|^p)|u|^{p-2}u+|u|^{2_{s}^{*}-2}u, \quad x\in \mathbb{R}^N, \end{align*} $under constraint$ \int_{\mathbb{R}^N}|u|^2dx c^2, $where $ N\ge2, \; I_{\alpha} \frac{1}{|x|^{N-\alpha}}, \alpha\in(N-2s, N), 2\le p<1+\frac{\...

Let f $f$ be a noncommutative polynomial of degree m ⩾ 1 $m\geqslant 1$ over an algebraically closed field F $F$ characteristic 0. If n − $n\geqslant m-1$ and α , 2 3 $\alpha _1,\alpha _2,\alpha _3$ are nonzero elements from such that + = 0 _1+\alpha _2+\alpha _3=0$ then every trace zero × $n\times n$ matrix can written as A _1 A_1+\alpha _2A_2+\alpha _3A_3$ for some i $A_i$ in the image M ( ) ...

Let$mathbb{C}^n$ be the space of $n$ complex variables. Let$Omega_{n,p_2,ldots,p_n}$ be a complete Reinhardt on$mathbb{C}^n$. The Minkowski functional on complete Reinhardt$Omega_{n,p_2,ldots,p_n}$ is denoted by $rho(z)$. The concept ofspirallike mapping of type $beta$ and order $alpha$ is defined.So, the concept of the strong and almost spirallike mappings o...

For a positive real number $\alpha$, let $\mathbb{N}_0[\alpha,\alpha^{-1}]$ be the semiring of all numbers $f(\alpha)$ for $f(x)$ lying in $\mathbb{N}_0[x,x^{-1}]$, which is Laurent polynomials over set nonnegative integers $\mathbb{N}_0$. In this paper, we study various factorization properties additive structure $\mathbb{N}_0[\alpha, \alpha^{-1}]$. We characterize when \alpha^{-1}]$ atomic. T...

A function $F(s)=\sum_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with $0\le\lambda_n\uparrow+\infty$ is called the Hadamard composition of genus $m\ge 1$ functions $F_j(s)=\sum_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where$P(x_1,...,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}x_1^{k_1}\cdot...\cdot x_p^{k_p}$ a homogeneous polynomial degree 1$. Let $M(\sigma,F)=\s...

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...