On the Derivative of a Polynomial

International Journal of Nonlinear Analysis and Applications, 2015

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.

2007

Let P(z) be a polynomial of degree n and M(P,t) = Max|z|=t |P(z)|. In this paper we shall estimate M(P 0 , ) in terms of M(P,r) where P(z) does not vanish in the disk |z| K, K 1, 0 r < < K and obtain an interesting refinement of some result of Dewan and Malik. We shall also obtain

Journal of Approximation Theory, 1990

Proceedings of The American Mathematical Society, 1974

Let !?",t denote the class of all polynomials p"(z) of degree at most « in z which satisfy ïanx\,\_ùpn(z)\ = 1, and |/>"(1)| =b, Q^b<l. Let c£(0, «], and set Mc> ») = sup I min \pn(z)\I. Upper estimates for nb(c, «) are obtained. Let Udenote the open unit disc in the complex z plane, Fits boundary, and let ^n-0 denote the class of all polynomials pn(z) of degree at most « in z, satisfying max2er|/?"(z)| = l and pn(l)=0. The extremal problem in question is to estimate fi(c,n) = sup min \pn(z)\ , Pneä'n 0\\z\=l-c/n I where 0<c^«. This problem was mentioned by Professor Paul Erdös during a lecture at the University of Montreal in July, 1971. He attributed the problem to G. Halász, of the Mathematical Institute of the Hungarian Academy of Sciences; Erdös asked if there exists a constant c such that fi(c, «)= 1-sn where £"->-0 as «->-co. It is easily seen that no such constant c exists. In fact, if pn e £?n¡a, then also qneSPn<s, where qn(z)=znpn(\/z), and by S. Bernstein's theorem [3, p. 45] on the derivative of a polynomial, \q'n(z)\^n for zeT. Hence it follows that \zn~xq'n(\\z)\-^n for z £ Fand by the maximum principle, also for all zeU. Replacing z by 1/z we find that \q'n(z)\ ^ «Izl""1 forall|z|^l.

Iranian Journal of Mathematical Sciences and Informatics, 2017

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.

Journal of Inequalities and Applications, 2011

For a polynomial p(z) of degree n, having all zeros in |z| ≤ 1, Jain is shown that

Lobachevskii Journal of Mathematics, 2012

Let p(z) be a polynomial of degree n. In this paper, we prove a more general results concerning maximum modulus of polynomials. Our results not only generalize some known polynomial inequalities but also improve the results due to Jain [6], Aziz and Rather .

Functiones et Approximatio Commentarii Mathematici

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by E n the set of the integer-valued polynomials with degree ≤ n, we show that the smallest positive integer c n satisfying the property: ∀P ∈ E n , c n P ′ ∈ E n is c n = lcm(1, 2,. .. , n). As an application, we deduce an easy proof of the well-known inequality lcm(1, 2,. .. , n) ≥ 2 n−1 (∀n ≥ 1). In the second part of the paper, we generalize our result for the derivative of a given order k and then we give two divisibility properties for the obtained numbers c n,k (generalizing the c n 's). Leaning on this study, we conclude the paper by determining, for a given natural number n, the smallest positive integer λ n satisfying the property: ∀P ∈ E n , ∀k ∈ N: λ n P (k) ∈ E n. In particular, we show that: λ n = p prime p ⌊ n p ⌋ (∀n ∈ N).

Journal of Approximation Theory, 1988

In this paper we consider the class of polynomials of the type p(z) = z s (a0 + ∑ n−s j=µ ajz j) , 1 ≤ µ ≤ n − s, 0 ≤ s ≤ n − 1 having some zeros at origin and rest of zeros on or outside the boundary of a prescribed disk, and obtain the generalization of well known results.

1998

Let P(z) be a polynomial of degree n with real or complex coefficients, In this paper we obtain a ring shaped region containing all the zeros of P( z), Our results include, as special cases, several known extensions of Enestrom-Kakeya theorem on the zeros of a polynomiaL We shall also obtain zero free regions for certain class of analytic functions. n THEOREMB. LetP(z) = L aki ¥=-°be apolynomial with complex coefficients k=O such that I arg ak -131 :::; a :::;~, k = 0, l, ... , n for some 13, and Mathematics subject classification (1991): 30CIO,30CI5.

Journal of Mathematical Analysis and Applications, 1980

Let P(z) be a polynomial of degree n with real or complex coefficients, In this paper we obtain a ring shaped region containing all the zeros of P( z), Our results include, as special cases, several known extensions of Enestrom-Kakeya theorem on the zeros of a polynomiaL We shall also obtain zero free regions for certain class of analytic functions. n THEOREMB. LetP(z) = L aki ¥=-°be apolynomial with complex coefficients k=O such that I arg ak -131 :::; a :::;~, k = 0, l, ... , n for some 13, and Mathematics subject classification (1991): 30CIO,30CI5.

Publications de l'Institut Math?matique (Belgrade)

We extend Aziz and Mohammad's result that the zeros, of a polynomial P (z) = n j=0 a j z j , ta j a j−1 > 0, j = 2, 3,. .. , n for certain t (> 0), with moduli greater than t(n − 1)/n are simple, to polynomials with complex coefficients. Then we improve their result that the polynomial P (z), of degree n, with complex coefficients, does not vanish in the disc |z − ae iα | < a/(2n); a > 0, max |z|=a |P (z)| = |P (ae iα)|, for r < a < 2, r being the greatest positive root of the equation x n − 2x n−1 + 1 = 0, and finally obtained an upper bound, for moduli of all zeros of a polynomial, (better, in many cases, than those obtainable from many other known results).

2011

In this paper, we prove some compact generalizations of some well-known Bernstein type inequalities concerning the maximum modulus of a polynomial and its derivative in terms of maximum modulus of a polynomial on the unit circle. Besides, an inequality for self-inversive polynomials has also been obtained, which in particular gives some known inequalities for this class of polynomials. All the inequalities obtained are sharp.

BIBECHANA

In this paper we shall obtain some zero-free regions for the derivative of a polynomial.