极性的广义不等式多项式的导数外文翻译资料

剩余内容已隐藏，支付完成后下载完整资料

原文：

A generalized inequality for the polar derivative of a polynomial

Singh et al. Journal of Inequalities and Applications 2013, 2013:183

Gulshan Singh1*, Wali Mohammad Shah2 and Abdul Liman3

Abstract: In this paper, we extend a result recently proved by Liman et al. to the polar derivative of a polynomial and thereby obtain some more general results for polynomials with restricted zeros.

MSC: 30A10; 30C10; 30C15

Key words: polynomial; inequality; zeros; polar derivative

1 Introduction and statement of results

Let be a polynomial of degree n, and let be its derivative, then . (1)

Inequality (1) is a famous result due to Bernstein and is best possible with equality holding for the polynomial ,whereis a complex number.

If we restrict ourselves to a class of polynomials having no zeros in, then the above inequality can be sharpened. In fact, Erdouml;s conjectured and later Lax [1] proved that if ,in , then

(2)

As a refinement of (2), Aziz and Dawood [2] proved that if P(z) is a polynomial of degree n having no zeros in , then

(3)

As an improvement of (3), Dewan and Hans [3] proved that if P(z) is a polynomial of degree n having no zeros in , then for any beta; withand ,

(4)

Let denote the polar derivative of the polynomial P(z) of degree n with respect to , then

.

The polynomial is of degree at most n-1 and it generalizes the ordinary derivative in the sense that

As an extension of (1) to the polar derivative, Aziz and Shah ([4], Theorem 4 with k = 1) showed that if P(z) is a polynomial of degree n, then for every complex number with ,

(5)

Inequality (5) becomes equality for

If we divide the two sides of (5) by and let , we get inequality (1).Aziz and Shah [5] proved that if P(z) is a polynomial of degree n that does not vanish in, then for every complex number with ,

(6)

The estimate (6) is best possible with equality for . If we divide both sides of (6) by and make, we get inequality (3).

As an improvement and generalization to (6) and (4), Liman et al. [6] recently proved the following theorem.

Theorem 1 If P(z) is a polynomial of degree n that does not vanish in , then for every complex number , with, and,

（7）

In this paper, we prove the following more general result which is an extension as well as generalization of Theorem and yields a number of known polynomial inequalities.

Theorem 2 Let P(z) be a polynomial of degree n that does not vanish in |z| lt; k, k le;1, then for all real or complex numbers with ||ge;k, kle;1, i = 1, 2, . . . , t, t le;n–1 and for any real or complex numberwith ||le;1 and for |z| =1,

Remark Theorem 1 is a special case of Theorem 2 when we take t = k =1.

If we take t =1 in Theorem 2, we get the following corollary.

Corollary 1 If P(z) is a polynomial of degree n that does not vanish in | z| lt; k, k le;1, then for all complex numbers , with ||ge;k, k le;1, ||le;1, and for | z| =1,

If we take k = 1 in Theorem 2, we get the following result.

Corollary 2 Let P(z) be a polynomial of degree n that does not vanish in | z|le;1, then for all real or complex numbers with ||ge;1, i =1,2,..., t, t le;n-1 and for any real or complex number beta; with | beta; |le;1 and for | z| = 1,

For beta;=0 and t =1 in Theorem 2, we get the following.

Corollary 3 Let P(z) be a polynomial of degree n that does not vanish in | z| lt; k, kle;1, then for any real or complex numberwith ||ge;k, kle;1,

If we take k =1 in Corollary 3, then (10) reduces to (6).

By taking t =1 in (8), dividing both sides of (8) by |alpha;| and letting |alpha;| → infin;, we have the following generalization of inequality (4).

Corollary 4 Let P( z) b e a polynomial of degree n that does not vanish in |z|lt; k, kle;1, then for any real or complex number beta; with |beta;|le;1 and |z| =1,

Taking beta;=1 and k=1 in Corollary 4, (11) reduces to (3).

2 Lemmas

We require the following lemmas. The first lemma follows from Laguerrersquo;s theorem [7,P.52] (see also [8]).

Lemma 1 If all the zeros of the nth degree polynomial P(z) lie in a circular region C, and if xi; is any zero of

the polar derivative of P(z), then both points xi; and alpha; may not lie outside of C.

By repeated applications of Lemma 1, we get the following result, when the circular region C is the circle |z|le;r.

Lemma 2 If all the zeros of the nth degree polynomial P(z) lie in |z|le;r and if none of the pointslie in |z|le;r, then each of the polar derivatives,t = 1, 2, . . . , n-1, has all its zeros in |z|le;r.

Lemma 3Ifis a polynomial of degree n having no zeros in the disk |z|lt; k, kge;1, then

The above lemma is due to Chan and Malik [9].

Lemma 4 Ifis a polynomial of degree n having all its zeros in the disk |z|le;kle;1, then

Proof of Lemma 4 Since all the zeros of P(z) lie in |z|le;kle;1, therefore all the zeros of lie in |z|ge;kge;1. Hence applying Lemma 3 to the polynomial ，we get

Hence, inequality (12) follows.

Lemma 5 Let ,be a polynomial of degree n having all its zeros in the disk |z|le;k, kle;1, then for every real or complex number alpha; with |alpha;| ge;k,k le;1 and for |z| =1, 

Proof of Lemma 5 Let , we have . Then it can be easily verified that

Since P(z) has all

剩余内容已隐藏，支付完成后下载完整资料

资料编号：[287159]，资料为PDF文档或Word文档，PDF文档可免费转换为Word