on the maximum modulus of 
on 
for arbitrary real or complex numbers
, 
with
, 
and
, and present certain sharp operator preserving inequa-lities between polynomials.
1. Introduction to the Statement of Results
Let 
denote the space of all complex polynomials 
of degree n. If
, then concerning the estimate of the maximum of 
on the unit circle 
and the estimate of the maximum of 
on a larger circle
, we have
(1)
and
(2)
Inequality (1) is an immediate consequence of S. Bernstein’s theorem (see [1-3]) on the derivative of a trigonometric polynomial. Inequality (2) is a simple deduction from the maximum modulus principle (see [4, p. 346] or [5, p. 158]). If we restrict ourselves to the class of polynomials 
having no zero in
, then (1) and (2) can be replaced by
(3)
and
(4)
Inequality (3) was conjectured by Erdös and later verified by Lax [6]. Ankeny and Rivlin [7] used Inequality (3) to prove Inequality (4).
As a compact generalization of Inequalities (1) and (2), Aziz and Rather [8] have shown that if
, then for every real or complex number 
with
, 
and
,
(5)
The result is sharp.
As a corresponding compact generalization of Inequalities (3) and (4), they [8] have also shown that if
, and 
for
, then for every real or complex number 
with
, 
,
(6)
for
. The result is sharp and equality in (6) holds for
,
.
Consider an operator B which carries a polynomial 
into
(7)
where
, 
and 
are such that all the zeros of
(8)
lie in the half plane
(9)
As a generalization of the Inequalities (1) and (2), Q.I. Rahman [9] proved that if
, then for
,
(10)
and if 
for
, then for
,
(11)
(see [9], Inequality (5.2) and (5.3)).
In this paper, we consider a problem of investigating the dependence of
on the maximum modulus of 
on 
for arbitrary real or complex numbers
, 
with
, 
and
, and develop a unified method for arriving at these results. In this direction we first present the following interesting result which is compact generalization of the Inequalities (1), (2), (5) and (10).
Theorem 1. If
, then for arbitrary real or complex numbers 
and 
with
, 
and
,
(12)
where
The result is best possible and equality in (12) holds for 
Remark 1. For 
from Inequality (12), we have for
, 
, 
and 
(13)
Remark 2. For 
and
, Inequality (12) reduces to
(14)
for
, 
and
, which contains Inequality (10) as a special case.
Remark 3. For
, Inequality (12) yields,
(15)
for 
and
.
If we choose 
in (12) and noting that all the zeros of 
defined by (8) lie in the half plane (9), we get:
Corollary 1. If
, then for all real or complex numbers 
and 
with
, 
, 
and
,
(16)
where 
is defined as in Theorem 1. The result is sharp and equality in (16) holds for
, 
For the case
, from (12) we obatin for all real or complex numbers 
and 
with
, 
, 
and
,
(17)
Inequality (17) is equivalent to the Inequality (5) for 
and
. For 
and
, Inequality (17) includes Inequality (2) as a special case.
Next we use Theorem 1 to prove the following result.
Theorem 2. If
, then for arbitrary real or complex numbers 
and 
with
, 
, 
and
,
(18)
where 
and 
is defined as in Theorem 1.
The result is sharp and equality in (18) holds for
, 
Remark 4. Theorem 2 includes some well known polynomial inequalities as special cases. For example, inequality (18) reduces to a result due to Q. I. Rahman ([8], Inequality (5.2) with 
and
). Also for
, Inequality (18) gives
(19)
for
, 
and
.
If we take 
in (18), we get:
Corollary 2. If
, then for all real or complex numbers 
and 
with
, 
, 
and
,
(20)
where 
is defined as in Theorem 1. The result is sharp and equality in (20) holds for
, 
For 
and
, 
, Theorem 2 includes a result due to A. Aziz and Rather [2] as a special case.
Inequality (12) can be sharpened if we restrict ourselves to the class of polynomials 
having no zeros in
. In this direction we next prove the following result which is a compact generalization of the Inequalities (3), (4) and (6).
Theorem 3. If 
and 
for
, then for arbitrary real or complex numbers 
and 
with
, 
, 
and
,
(21)
where 
is defined as in Theorem 1. The result is sharp and equality in (21) holds for 
Remark 5. Inequality (11) is a special case of the Inequality (21) for 
and
. If we choose 
in (21) and note that all the zeros of 
defined by (8) lie in the half plane defined by (9), it follows that if 
and 
for
, then for
, 
and
, 
,
(22)
Setting 
in (22), we obtain for
,
(23)
for
, 
and
.
Taking 
in (22), we obtain for
, 
and
,
(24)
which in particular gives Inequality (3).
Next choosing 
in (21), we immediately get for
, 
and
, 
,
(25)
which is a compact generalization of the Inequalities (3), (4) and (6). The result is sharp and equality in (25) holds for
, 
If we put 
in (25), we get the following result.
Corollary 3. If
, and 
for
, then for every real or complex number 
with
, 
and
,
(26)
A polynomial 
is said to be self-inversive if
where
. It is known [610] that if 
is a self-inversive polynomial, then
(27)
Here finally, we establish the following result for self-inversive polynomials Theorem 4. If 
is a self-inversive polynomial, then for arbitrary real or complex numbers 
and 
with
, 
, 
and
,
(28)
where 
is defined as in Theorem 1. The result is sharp and equality in (21) holds for 
The following result is an immediate consequence of Theorem 4.
Corollary 4. If 
is a self-inversive polynomial, then for arbitrary real or complex numbers 
and 
with
, 
, 
and
,
(29)
where 
is defined as in Theorem 1. The result is best possible For 
the Inequality (29) reduces to
(30)
Remark 6. Inequality (6) is a special case of the Inequality (30). Many other interesting results can be deduced from Theorem 4 in the same way as we have deduced from Theorem 1 and Theorem.
2. Lemmas
For the proofs of these theorems, we need the following lemmas. The first lemma can be easily proved.
Lemma 1. If 
and 
has all its zeros in
, then for every 
and
,
(31)
The next Lemma follows from corollary 18.3 of [11, p. 65].
Lemma 2. If 
and 
has all its zeros in
, then all the zeros of 
also lie in
.
Lemma 3. If 
and 
does not vanish in
, then for arbitrary real or complex numbers 
and 
with
, 
, 
and
,
(32)
where
and 
is defined as in Theorem 1.
The result is sharp and equality in (32) holds for 
Proof of Lemma 3. Since the polynomial 
has all its zeros in 
for every real or complex number 
with
, the polynomial
where
, has all its zeros in 
with atleast one zero in
, so that we can write
where 
and 
is a polynomial of degree 
having all its zeros in
.
Applying lemma 1 to the polynomial
, we obtain for 
and
,
This implies for 
and
,
(33)
Since 
so that 
for 
and
, from Inequality (33), we obtain for 
and
,
(34)
Equivalently,
for 
and
. Hence for every real or complex number 
with 
and 
we have
(35)
for
. Also, Inequality (34) can be written as
(36)
for every 
and 
Since 
and
, from inequality (36), we obtain for 
and
,
Equivalently,
Since all the zeros of 
lie in
, a direct application of Rouche’s theorem shows that the polynomial 
has all its zeros in 
for every real or complex number 
with
. Applying Rouche’s theorem again, it follows from (35) that for arbitrary real or complex numbers 
with
, 
and
, all the zeros of the polynomial
lie in 
with
. Applying Lemma 2 to the polynomial 
and noting that B is a linear operator, it follows that all the zeros of the polynomial
lie in 
for all real or complex numbers 
with
, 
, 
and
. This implies
(37)
for
, 
, 
and
. If Inequality (38) is not true, then there is a point 
with 
such that
But all the zeros of 
lie in
, therefore, it follows (as in case of
) that all the zeros of
lie in
. Hence by Lemma 2, all the zeros of
lie in
, so that
We take
then 
is a well defined real or complex number with 
and with this choice of
, from (37) we obtain 
where
. This contradicts the fact that all the zeros of 
lie in
. Thus
for
, 
, 
and
. This proves (38) and hence Lemma 3.
3. Proofs of the Theorems
Proof of Theorem 1. Let
, then
for
. By Rouche’s Theorem, it follows that all the zeros of the polynomial 
lie in 
for every real or complex number 
with
, therefore, as before (as in Lemma 3), we conclude that all the zeros of the polynomial
lie in 
for all real or complex numbers 
and 
with 
and
. Hence by Lemma 2, the polynomial
has all its zeros in 
for every real or complex number 
with
. This implies for every real or complex numbers 
and 
with
, 
and
,
(38)
If Inequality (40) is not true, then there is a point 
with 
such that
Since
, we take
so that 
is a well defined real or complex number with 
and with this choice of
, from (39) we get 
where
. This contradicts the fact that all the zeros of 
lie in
. Thus for every real or complex numbers 
and 
with
, 
and
,
This completes the proof of Theorem 1.
Proof of Theorem 2. Let
, then
for
. If 
is any real or complex number with
, then by Rouche’s Theorem, the polynomial 
does not vanish in
. Applying Lemma 3 to the polynomial 
and using the fact that B is a linear operator, it follows that for all real or complex numbers 
and 
with
, 
, 
and for 
where
Using the fact that
, we obtain
for all real or complex numbers 
and 
with
, 
, 
and
. Now choosing the argument of 
such that
which is possible by Theorem 1, we get
for
, 
, 
and
. This implies
for
, 
, 
and
. Letting
, we obtain
which is inequality (18) and this proves Theorem 2.
Proof of Theorem 3. Lemma 3 and Theorem 2 together yields for all real or complex numbers 
and 
with
, 
, 
and
,
which gives
which is the Inequality (21) and this completes the proof of Theorem 3.
Proof of Theorem 4. Since 
is a self-inversive polynomial of degree n, therefore
for all
. This implies, in particular, that for all real or complex numbers 
and 
with
, 
, 
and
,
Combining this with Theorem 2, the desired result follows immediately. This completes the proof of Theorem 4.
4. Acknowledgements
Authors are thankful to the referee for his suggestions.