Some Inequalities on Polar Derivative of Polynomial Having No Zero in a Disc ()
In this paper, we first extend the above inequality to polar derivative of a polynomial. Further, as an application of our result, we extend a result due to Dewan et al. [Southeast Asian Bull. Math., 27 (2003), 591-597] to polar derivative.
Keywords:
1. Introduction and Statement of Results
Let
be a polynomial of degree n. Then according to the well-known Bernstein’s inequality [1] .
. (1.1)
Equality holds in (1.1) if and only if
has all its zeros at the origin.
If we restrict ourselves to the class of polynomials having no zero in
, then inequality (1.1) can be sharpened. It was conjectured by Erdös and later verified by Lax [2] that if
in
, then (1.1) can be replaced by
. (1.2)
Inequality (1.2) is best possible and equality attains for
,
.
Malik [3] extended (1.2) by considering the class of polynomials
of degree n not vanishing in
,
, and proved
. (1.3)
Qazi [4] considered a more general class of polynomials
,
, having no zero in
,
, and obtained the following, which is a generalization as well as an improvement of (1.3).
Theorem A. If
,
, is a polynomial of degree n having no zero in
, k ≥ 1, then
. (1.4)
Inequality (1.4) is sharp and equality holds for the polynomial
where n is a multiple of
.
By involving
, the above theorem was improved by Dewan et al. [5] for
.
Theorem B. If
,
, is a polynomial of degree n having no zero in
, k ≥ 1, then
(1.5)
Inequality (1.5) is best possible for
where n is a multiple of
with
.
Remark 1. Theorem B proved by Dewan et al. [5] seems to have a deficiency in the sense that for
the corresponding result was not specified. In fact, by simple calculation, we find the result to be the equality
. (1.6)
Let
be a polynomial of degree n and
be any real or complex number, then the polar derivative of
, denoted by
, is defined as
. (1.7)
The polynomial
is of degree at most
and it generalizes the ordinary derivative
of
in the sense that
.
The polynomial
is called by Laguerre ( [6] , p. 48) the “émanant” of
, by Pólya and Szegö [7] the “derivative of
with respect to the point
” and by Marden ( [8] , p. 44) simply “ the polar derivative of
”.
Aziz [9] extended (1.3) to the polar derivative of
by showing that if
has no zero in
,
, then for every real or complex number
with
,
. (1.8)
Inequality (1.8) is best possible and equality holds for
with
and
.
Further, by considering a more general class of polynomials
,
, of degree n
having no zero in
,
, then for every real or complex number
with
, it was Dewan and Singh [10] who proved the following inequality which generalizes inequality (1.8) due to Aziz [9] .
. (1.9)
In this paper, we first extend Theorem A to polar derivative of a polynomial, which gives an improvement of (1.9). More precisely, we prove.
Theorem 1. If
,
, is a polynomial of degree n having no zero in
, k ≥ 1, then for every real or complex number
with
,
(1.10)
Equality in (1.10) holds for
with
, extremal polynomial being
,
.
Remark 2. To prove that the bound of Theorem 1is better than that of (1.9), it is sufficient to prove that
,
i.e. equivalently,
![]()
i.e.
![]()
which is true since
,
, and by (2.5) of Lemma 2.3, i.e.,
.
Further, if we put
in Theorem 1, we get the following result which is an improvement of inequality (1.8) due to Aziz [9] .
Corollary 1. If
is a polynomial of degree n having no zero in
,
, then for every real or complex number
with
,
. (1.13)
Remark 3. Inequality (1.13) is the corresponding polar derivative version of a result proved by Govil et al. ( [11] , Inequality (10)).
Remark 4. As mentioned earlier, inequality (1.13) improves inequality (1.8) and is evident from Remark 2, for the paticularcase
.
It is of interest that as an application of Theorem 1, we have been able to obtain an independent proof of a re-
sult proved by Mir and Dar ( [12] , Theorem 1), which involves
and extends Theorem B to polar de-
rivative which also improves upon Theorem 1 for
. In fact, we prove
Theorem 2. If
,
, is a polynomial of degree n having no zero in
, k ≥ 1, then for every real or complex number
with
,
(1.11)
(1.12)
Equality occurs in (1.11) for
with
, extremal polynomial being
,
.
If we divide both sides of the above inequalities (1.11) and (1.12) by
and make
, we obtain the inequalities (1.5) and (1.6) respectively.
Remark 5. For
, Theorem 2 gives the following
Corollary 2. If
is a polynomial of degree n having no zero in
,
, then for every real or complex number
with
,
(1.14)
Inequality (1.14) is best possible for
, with
and
.
Remark 6. It is obvious that Corollary 2 is an improvement of Corollary 1.
2. Lemmas
The following lemmas are required in the proofs of the theorems.
Lemma 2.1. If
is a polynomial of degree n, then on
,
, (2.1)
where
.
The above lemma is a special case of a result due to Govil and Rahman [13] .
Lemma 2.2. If
is a polynomial of degree n, then for every real or complex number
, we have on
,
. (2.2)
Proof of Lemma 2.2. The proof of this lemma is simple and follows as a part ( [10] , proof of Theorem 1), but for the sake of completeness, we outline it. Let
. Then it is easy to verify that on
,
. (2.3)
Now, for every real or complex number
, the polar derivative of
with respect to
is
. (2.4)
This implies on
,
![]()
which completes the proof of Lemma 2.2.
Lemma 2.3. If
,
, is a polynomial of degree n having no zero in
, k ≥ 1, then
(2.4)
and
(2.5)
Lemma 2.3 is due to Qazi ( [4] , Proof and Remark of Lemma 1).
3. Proofs of the Theorems
Proof of Theorem 1. On
, by Lemma 2.1, we have
(3.1)
and by inequality (2.4) of Lemma 2.3, we have
. (3.2)
Combining (3.1) and (3.2), we obtain for
,
,
which gives for
,
.
Now, if
, then multiplying both sides of the above inequality by
, we get
. (3.3)
Inequality (3.3) when combined with Lemma 2.2, gives for
and
,
![]()
which is equivalent to
,
from which Theorem1 follows.
Proof of Theorem 2. First, we prove inequality (1.11).
Let
. Since
has no zero in
, k ≥ 1 the polynomial
has no zero in
, k ≥ 1, for every real or complex number
with
. The claim is obvious if
has a zero on
for then
and hence
. If
has no zero on
, then we have
on
and the claim follows from Rouché’s theorem. Thus, in any case
has no zero in
, k ≥ 1 and therefore on applying Theorem 1 to the polynomial
, that is to
, where
, we have for every real or complex number
with
,
![]()
which implies
(3.4)
Let
be a point on the unit circle such that
, then (3.4), in particular, gives
(3.5)
Now, we choose the argument of
in (3.5) such that
.
Then (3.5) becomes
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Finally, making
in the above inequality, we obtain inequality (1.11).
For
, the polynomial is simply
having no zero in
,
. As
has no
zero in
, therefore
. Then
,
, and
. From these three equations, equality (1.12) follows readily.