let $p(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. in this paper, some algebraic and geometrical properties of the $k$-numerical range of $p(lambda)$ are investigated. in particular, the relationship between the $k$-numerical range of $p(lambda)$ and the $k$-numerical range of its companion linearization is stated. moreover, the $k$-numerical ...

for a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, dewan et al [k. k. dewan, n. singh and a. mir, extension of some polynomial inequalities to the polar derivative, j. math. anal. appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). in this paper we improve and extend the above inequality. our result generalizes certai...

rice (oryza sativa l.) is one of the iran&apos;s most strategic crops with water shortage as the most important abiotic stress limiting its yield. in the present research the effects of: variety, water regime, nitrogen (n) and phosphorus (p) levels on n, p and k uptake were studied under greenhouse conditions. the experiment design was a factorial one based upon a randomized complete blocks of ...

Let $P$ be a complex polynomial of the form $P(z)=zdisplaystyleprod_{k=1}^{n-1}(z-z_{k})$,where $|z_k|ge 1,1le kle n-1$ then $ P^prime(z)ne 0$. If $|z|

‎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|=...