The Journal of the Indian Mathematical Society

V. K. Jain ¹

Affiliations
1 Mathematics Department, I. I. T., Kharagpur - 721 302, IN

Abstract

Using one of Arestov's integral inequalities for polynomials we have obtained a generalization of Rahman and Schmeisser's result on polynomial inequalities and using Arestov's another integral inequality we have obtained a generalization of previously used Arestov's inequality. Certain other related results have also been suggested.

Keywords

Generalization, Integral Inequalities, Polynomials with Zeros on or Outside the Unit Circle, Polynomials with Zeros in or on the Unit Circle.

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V. K. Jain ¹

Affiliations
1 Mathematics Department, Indian Institute of Technology, Kharagpur - 721302, IN

Abstract

For an arbitrary polynomial p(z), let M(p, r)=max_|z|=_r|p(z)|. For a self-inversive polynomial p(z) (with respect to the unit circle), of degree n it is known that

M(p,1)=n/₂ M(p, 1).

By considering a self-inversive polynomial p(z) (with respect to the circle |z|=k, (k>0)), of degree n we have obtained

M(p',1)≥{ n/(1+kⁿ)} M(p,1), k≥1, {n/(1 + k²⁻ⁿ)}M(p,1), k < 1

M(p',1) {n/(1+ k²⁻ⁿ)} M(p,1), k≥1; {n/(1+kⁿ)}M(p,1), k < 1.

( provided p′(z) and p′(k²z) attain their maximum moduli on the unit circle, at the same point ) ; a generalization of this result.

Keywords

Self-Inversive Polynomial, Zeros, Maximum Modulus.

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V. K. Jain ¹

Affiliations
1 Mathematics Department, I.I.T. Kharagpur-721 302, IN

Abstract

It is known that the polynomial p(z) = Σa_jz^j of degree n, with complex coefficients, (i) has all its zeros in the disc

                                 |z|≤R.

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V. K. Jain ¹

Affiliations
1 Mathematics Department, I.I.T., Kharagpur-721302, IN

Abstract

Let f(z) be an arbitrary entire function and M(f,r)=max |f(z)|. If p(z) is a polynomial of degree n, having no zeros in |zl<k≤1, with M(p,1)=1, then for k=1, it is known that.
M(p,R)≤Rⁿ+1/2, R>1.

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V. K. Jain ¹

Affiliations
1 Department of Mathematics, I.I.T., Kharagpur-721302, IN

Abstract

For a polynomial P(z)=Σa_jz^j+zⁿ, of degree n(≥3), with complex coefficients, where not all of the numbers a₀, a₁,...,a_n-s(s≥3), are equal to zero, a closed circular disc. where containing all its zeros has been obtained. The result is best possible and gives, in many cases, better upper bounds (for moduli of zeros of P(z)), than those obtainable by other results of similar type.

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V. K. Jain ¹

Affiliations
1 Mathematics Department, l.l.T., Kharagpur-721 302, IN

Abstract

Certain Sharp Inequalities for Polynomial a₀+a₁Z+a₂Z²+...+a_n-1Z^n-1+a_nZⁿ with |a₀|=|a_n|,|a₁|=|a_n-1|,|a₂| = |a_n-2|,...

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V. K. Jain ¹

Affiliations
1 Mathematics Department, I.I.T., Kharagpur, IN

Abstract

Let p(z) be a polynomial of degree n and M(p, r) = Max |p(z)|. In [4], we had considered the problem of obtaining the converse of the following result due to Aziz and Mohommad [2].

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