Inequalities on Polynomial
نویسنده
چکیده
We give explicit bounds for the absolute values of the coefficients of the divisors of a complex polynomial. They are expressed in function of the coefficients and of upper and lower bounds for the roots. These bounds are compared with other estimates, in particular with the inequality of Beauzamy [B. Beauzamy, Products of polynomials and a priori estimates for coefficients in polynomial decompositions: A sharp result, J. Symbolic Comput., 13 (1992), 463–472]. Through examples it is proved that for some cases our evaluations give better upper limits.
منابع مشابه
On Bernstein Type Inequalities for Complex Polynomial
In this paper, we establish some Bernstein type inequalities for the complex polynomial. Our results constitute generalizations and refinements of some well-known polynomial inequalities.
متن کاملOn the polar derivative of a polynomial
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...
متن کاملL$^q$ inequalities for the ${s^{th}}$ derivative of a polynomial
Let $f(z)$ be an analytic function on the unit disk ${zinmathbb{C}, |z|leq 1}$, for each $q>0$, the $|f|_{q}$ is defined as followsbegin{align*}begin{split}&left|fright|_q:=left{frac{1}{2pi}int_0^{2pi}left|f(e^{itheta})right|^qdthetaright}^{1/q}, 0
متن کاملInequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin
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|=...
متن کاملOn the $s^{th}$ derivative of a polynomial
For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.
متن کاملOn the $s^{th}$ Derivative of a Polynomial-II
The paper presents an $L^{r}-$ analogue of an inequality regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle of arbitrary radius but greater or equal to one. Our result provides improvements and generalizations of some well-known polynomial inequalities.
متن کامل