Improved Polar Extensions of an Inequality for a Complex Polynomial with All Zeros on a Circle

Abstract

If $f\left( w \right)$ is a polynomial of degree $m$ having all its zeros on $\left| w \right| = \rho$, $\rho \le 1$, then Govil proved that

$\mathop {\max }\limits_{{\left| {\varvec{w}} \right| = 1}} \left| {\user2{f^{\prime}}\left( {\varvec{w}} \right)} \right| \le \frac{{\varvec{m}}}{{{\varvec{\rho}}^{{\varvec{m}}} + {\varvec{\rho}}^{{{\varvec{m}} - 1}} }}\mathop {\max }\limits_{{\left| {\varvec{w}} \right| = 1}} \left| {{\varvec{f}}\left( {\varvec{w}} \right)} \right|.$

In this paper, we proved refined extensions of this inequality in polar derivative under the same hypothesis.

1 Introduction

Let $f\left( w \right) = \sum\nolimits_{j = 0}^{m} {c_{j} w^{j} }$ be a polynomial of degree $m$ and let

$${\text{Max}}\left( {f,r} \right) = \mathop {\max }\limits_{\left| w \right| = r} \left| {f\left( w \right)} \right|.$$

If $f\left( w \right)$ has no zero in $\left| w \right| < \rho ,$ $\rho \ge 1$, Malik [8] proved that

$${\text{Max}}\left( {f^{\prime},1} \right) \le \frac{m}{1 + \rho }{\text{Max}}\left( {f,1} \right),$$

(1)

for which the equality holds for the polynomial $\left( {w + \rho } \right)^{m}$.

A natural question that arise is whether there exists an analogous inequality of (1) for $f\left( w \right)$ having no zero in $\left| w \right| < \rho$, $\rho \le 1$. In this regard, Govil [5, 6] proved the following two results.

Theorem 1

[5] If $f\left( w \right)$ is a polynomial of degree $m$ having no zero in $\left| w \right| < \rho$, $\rho \le 1$, then

$${\text{Max}}\left( {f^{\prime},1} \right) \le \frac{m}{{1 + \rho^{m} }}{\text{Max}}\left( {f,1} \right),$$

(2)

provided that $\left| {f^{\prime}\left( w \right)} \right|$ and $\left| {F^{\prime}\left( w \right)} \right|$ attain their maxima at the same point on the circle $\left| w \right| = 1$, where

$$F\left( w \right) = w^{m} \overline{{f\left( {\frac{1}{{\overline{w}}}} \right)}} .$$

Theorem 2

[6] If $f\left( w \right)$ is a polynomial of degree $m$ having all its zeros on $\left| w \right| = \rho ,\rho \le 1,$

$${\text{Max}}\left( {f^{\prime},1} \right) \le \frac{m}{{\rho^{m} + \rho^{m - 1} }}{\text{Max}}\left( {f,1} \right).$$

(3)

In literature we find refinements, generalizations and extensions of Theorems 1 and 2 by involving some coefficients of the polynomial $f\left( w \right)$ (see [2,3,4]).

Definition 1

Let $f\left( w \right)$ be a polynomial of degree $m$ and let $\beta$ be any complex number. The polar derivative of $f\left( w \right)$ with respect to the point $\beta$ , denoted by $D_{\beta } f\left( w \right)$ , is defined as

$$D_{\beta } f\left( w \right) = mf\left( w \right) + \left( {\beta - w} \right)f^{\prime}\left( w \right).$$

$D_{\beta } f\left( w \right)$ is a polynomial of degree at most $m - 1$. It can be considered as a generalized form of the ordinary derivative of $f\left( w \right)$ with respect to $w$ due to the fact that

$$\mathop {\lim }\limits_{\beta \to \infty } \frac{{D_{\beta } f\left( w \right)}}{\beta } = f^{\prime}\left( w \right).$$

Polar derivative extension of (1) was proved by Aziz [1], who under the same hypothesis on $f\left( w \right)$ proved that

$${\text{Max}}\left( {D_{\beta } f\left( w \right),1} \right) \le m\left( {\frac{\rho + \left| \beta \right|}{{1 + \rho }}} \right){\text{Max}}\left( {f,1} \right), \quad {\text{where}}\, \left| \beta \right| \ge 1.$$

(4)

2 Lemmas

We need the following results to prove our results.

The following lemma is due to Govil and Rahman [7].

Lemma 1

[7] If $f\left( w \right)$ is a polynomial of degree $m$ having all its zeros on $\left| w \right| = \rho$, $\rho \le 1$, then on $\left| w \right| = 1$.

$$\left| {f^{\prime}\left( w \right)} \right| + \left| {F^{\prime}\left( w \right)} \right| \le m \;{\text{Max}}\left( {f,1} \right).$$

(5)

The next two lemmas are due to Barchand et al. [3].

Lemma 2

[3] If $f\left( w \right) = \sum\nolimits_{j = 0}^{n} {c_{j} } w^{j}$ is a polynomial of degree $m$ having all its zeros in $\left| w \right| \le \rho ,$ $\rho \le 1$, then

$$\frac{1}{\rho m}\left( {\frac{{c_{m - j} }}{{c_{m} }}} \right) \le 1.$$

(6)

Lemma 3

[3] If $f\left( w \right) = \sum\nolimits_{j = 0}^{n} {c_{j} } w^{j}$ is a polynomial of degree $m$ having all its zeros on $\left| w \right| = \rho ,$ $\rho \le 1$, then on $\left| w \right| = 1$.

$${\text{Max}}\left( {f^{\prime},1} \right) \le \frac{m}{{\rho^{m} + \rho^{m - 1} }}E_{\rho } \;{\text{Max}}\left( {f,1} \right),$$

(7)

where

$$E_{\rho } = \frac{{\left( {1 + \left| t \right|} \right)\left( {\rho^{2} + \left| t \right|} \right) + \rho \left( {m - 1} \right)\left| {s - t^{2} } \right|}}{{\left( {1 - \left| t \right|} \right)\left( {1 - \rho + \rho^{2} + \rho \left| t \right|} \right) + \rho \left( {m - 1} \right)\left| {s - t^{2} } \right|}},$$

$$t = \frac{1}{\rho m}\left( {\frac{{\overline{c}_{m - 1} }}{{\overline{c}_{m} }}} \right)$$

$$s = \frac{2}{{\rho^{2} m\left( {m - 1} \right)}}\left( {\frac{{\overline{c}_{m - 2} }}{{\overline{c}_{m} }}} \right).$$

Lemma 3 is, in fact, a refinement of Theorem 2 due to Govil [6] (see Remark 3).

Remark 1

Under the hypothesis of Lemma 3, we have (see [3, Lemma 2.6]).

$$\left| t \right| = \frac{1}{\rho m}\left| {\frac{{\overline{c}_{m - 1} }}{{\overline{c}_{m} }}} \right| \le 1.$$

The following lemma was proved by Malik [8].

Lemma 4

[8] If $f\left( w \right)$ is a polynomial of degree $m$ having no zero in $\left| w \right| < \rho$, $\rho \ge 1$, then

$$\rho \left| {f^{\prime}\left( w \right)} \right| \le \left| {F^{\prime}\left( w \right)} \right|,$$

(8)

where $F\left( w \right) = w^{m} \overline{{f\left( {\frac{1}{{\overline{w}}}} \right)}} .$

Lemma 5

If $f\left( w \right)$ is a polynomial of degree $m$ having all its zeros on $\left| w \right| = \rho ,$ $\rho \le 1$, then on $\left| w \right| = 1$.

$$\left| {F^{\prime}\left( w \right)} \right| \le \rho \left| {f^{\prime}\left( w \right)} \right|.$$

(9)

Proof

Since $f\left( w \right)$ has all its zeros on $\left| w \right| = \rho$, $\rho \le 1$, then $F\left( w \right)$ has all its zeros on $\left| w \right| = \frac{1}{\rho }$, $\frac{1}{\rho } \ge 1$. This implies that $F\left( w \right)$ has no zeros in $\left| w \right| < \frac{1}{\rho }$, $\frac{1}{\rho } \ge 1.$ Thus, applying Lemma 1 to the polynomial $F\left( w \right)$ we get, on $\left| w \right| = 1$.

$${ }\frac{1}{\rho }\left| {F^{\prime}\left( w \right)} \right| \le \left| {f^{\prime}\left( w \right)} \right|.$$

$$\therefore \left| {F^{\prime}\left( w \right)} \right| \le \rho \left| {f^{\prime}\left( w \right)} \right|.$$

3 Main Results

In this paper, we prove polar extensions of Theorem 2. Precisely, we prove the following result.

Theorem 3

If $f\left( w \right) = \mathop \sum \limits_{j = 0}^{n} c_{j} w^{j}$ is a polynomial of degree $m$ having all its zeros on $\left| w \right| = \rho ,$ $\rho \le 1$, and $\beta \in {\mathbb{C}}$ with $\left| \beta \right| \ge 1$, then on $\left| w \right| = 1$.

• (a)
  $$\left| {D_{\beta } f\left( w \right)} \right| \le m\left( {1 + \frac{\left| \beta \right| - 1}{{\rho^{m} + \rho^{m - 1} }}E_{\rho } } \right){\text{Max}}\left( {f,1} \right).$$

  (10)
• (b)
  $$\left| {D_{\beta } f\left( w \right)} \right| \le m\left( {\frac{\left| \beta \right| + \rho }{{\rho^{m} + \rho^{m - 1} }}E_{\rho } } \right){\text{Max}}\left( {f,1} \right).$$

  (11)

where

$$E_{\rho } = \frac{{\left( {1 - \left| t \right|} \right)\left( {\rho^{2} + \left| t \right|} \right) + \rho \left( {m - 1} \right)\left| {s - t^{2} } \right|}}{{\left( {1 - \left| t \right|} \right)\left( {1 - \rho + \rho^{2} + \rho \left| t \right|} \right) + \rho \left( {m - 1} \right)\left| {s - t^{2} } \right|}},$$

(12)

$$t = \frac{1}{\rho m}\left( {\frac{{\overline{c}_{m - 1} }}{{\overline{c}_{m} }}} \right)$$

(13)

$$s = \frac{2}{{\rho^{2} m\left( {m - 1} \right)}}\left( {\frac{{\overline{c}_{m - 2} }}{{\overline{c}_{m} }}} \right).$$

(14)

Proof

Let $F\left( w \right) = w^{m} \overline{{f\left( {\frac{1}{{\overline{w}}}} \right)}}$. Then on $\left| w \right| = 1$.

$$\left| {F^{\prime}\left( w \right)} \right| = \left| {mf\left( w \right) - wf^{\prime}\left( w \right)} \right|.$$

(15)

For $\beta \in {\mathbb{C}}$, polar derivative of $f\left( w \right)$ with respect to $\beta$ is

$$D_{\beta } f\left( w \right) = mf\left( w \right) + \left( {\beta - w} \right)f^{\prime}\left( w \right).$$

Therefore,

$$\left| {D_{\beta } f\left( w \right)} \right| = \left| {mf\left( w \right) + \left( {\beta - w} \right)f^{\prime}\left( w \right)} \right|$$

$$\le \left| {mf\left( w \right) - wf^{\prime}\left( w \right)} \right| + \left| \beta \right|\left| {f^{\prime}\left( w \right)} \right|$$

(16)

$$= \left| {F^{\prime}\left( w \right)} \right| + \left| \beta \right|\left| {f^{\prime}\left( w \right)} \right|$$

(17)

$$= \left| {F^{\prime}\left( w \right)} \right| + \left| {f^{\prime}\left( w \right)} \right| + \left( {\left| \beta \right| - 1} \right)\left| {f^{\prime}\left( w \right)} \right|.$$

(18)

Using (5) of Lemma 1 in (18) we have on $\left| w \right| = 1$

$$\left| {D_{\beta } f\left( w \right)} \right| \le m{\text{Max}}\left( {f,1} \right) + \left( {\left| \beta \right| - 1} \right){\text{Max}}\left( {f^{\prime},1} \right).$$

(19)

Now using (7) of Lemma 3 in (19) we have for $\left| w \right| = 1$

$$\left| {D_{\beta } f\left( w \right)} \right| \le m\left( {1 + \frac{\left| \beta \right| - 1}{{\rho^{m} + \rho^{m - 1} }}E_{\rho } } \right){\text{Max}}\left( {f,1} \right),$$

which proves (a).

We now prove (b).

Using (9) of Lemma 5 in (17), we get for $\left| w \right| = 1$

$$\left| {D_{\beta } f\left( w \right)} \right|$$

$$\le \rho \left| {f^{\prime}\left( w \right)} \right| + \left| \beta \right|\left| {f^{\prime}\left( w \right)} \right|$$

(20)

$$\le \left( {\left| \beta \right| + \rho } \right){\text{Max}}\left( {f^{\prime},1} \right).$$

(21)

Using (7) of Lemma 3, we obtain for $\left| w \right| = 1$.

$$\left| {D_{\beta } f\left( w \right)} \right| \le m\left( {\frac{\left| \beta \right| + \rho }{{\rho^{m} + \rho^{m - 1} }}E_{\rho } } \right){\text{Max}}\left( {f,1} \right),$$

which proves (b).

Remark 2

Dividing inequalities (10) and (11) by $\left| \beta \right|$ and taking $\left| \beta \right| \to \infty$, both reduce to

$${\text{Max}}\left( {f^{\prime},1} \right) \le \frac{m}{{\rho^{m} + \rho^{m - 1} }}E_{\rho } {\text{Max}}\left( {f,1} \right),$$

which is the conclusion of Lemma 3, where $E_{\rho }$ is given by (12).

Remark 3

Inequalities (10) and (11) are improved extensions of inequality (3) to polar derivative. In other words, the ordinary form of (10) and (11) obtained in Remark 2 is an improvement of (3). To see this, it is sufficient to show that

$$E_{\rho } \le 1.$$

That is,

$$\frac{{\left( {1 - \left| t \right|} \right)\left( {\rho^{2} + \left| t \right|} \right) + \rho \left( {m - 1} \right)\left| {s - t^{2} } \right|}}{{\left( {1 - \left| t \right|} \right)\left( {1 - \rho + \rho^{2} + \rho \left| t \right|} \right) + \rho \left( {m - 1} \right)\left| {s - t^{2} } \right|}} \le 1.$$

i.e., $\rho^{2} + \left| t \right| \le 1 - \rho + \rho^{2} + \rho \left| t \right|$

which holds as $\rho \le 1$ and $\left| t \right| \le 1$ (by Remark 1).