Article
Kyungpook Mathematical Journal 2022; 62(3): 455-465
Published online September 30, 2022
Copyright © Kyungpook Mathematical Journal.
Remark on Some Recent Inequalities of a Polynomial and its Derivatives
Barchand Chanam∗, Khangembam Babina Devi and Thangjam Birkramjit Singh
Department of Mathematics, National Institute of Technology Manipur, Imphal, Langol, 795004, India
e-mail : barchand_2004@yahoo.co.in, khangembambabina@gmail.com and birkramth@gmail.com
Received: December 6, 2020; Revised: October 12, 2021; Accepted: November 8, 2021
Abstract
We point out a flaw in a result proved by Singh and Shah [Kyungpook Math. J., 57(2017), 537-543] which was recently published in Kyungpook Mathematical Journal. Further, we point out an error in another result of the same paper which we correct and obtain integral extension of the corrected form.
Keywords: polynomials, s-fold zero, inequality, integral extension
1. Introduction and Statement of Results
Let
From a well known fact of analysis ([12],[14]), we know
Thus, it is appropriate to denote
Further, if we define
If
Inequality (1.4) is known in the literature as Bernstein's Inequality [3] and it is best possible with equality holding for the polynomial
If
Inequality (1.5) was conjectured by Erdös which was later proved by Lax [7]. Inequality (1.5) is best possible and become equality for polynomials which have all the zeros on
Under the same hypothesis on
Equality holds in (1.6) for
Malik [8] generalized (1.5) for polynomial
Chan and Malik [4] considered more general lacunary type of polynomials
Theorem A. Let
Equality occurs in (1.8) for
As an improvement of Theorem A, Pukhta [9] proved
Theorem B. Let
Equality in (1.9) occurs for
Singh and Shah [13] proved the following result which apparently is a generalization and an improvement of Theorems A and B for the class of polynomials with
Theorem C. Let
In the same paper [13], the authors further proved the following result as generalization of Theorem C.
Theorem D. Let
2. Lemmas
We need the following lemmas to prove the theorem.
Lemma 2.1. If
and
where
This lemma is due to Qazi [10].
If
where
This lemma is due to Gardner et al. [6].
Lemma 2.3. The function
is a non-decreasing function of
This lemma is due to Gardner et al. [6, Lemma 2.6]. However, the authors did not define the quantities
From (2.3), we can show that
which implies that, for
Lemma 2.4. If
The above lemma was proved by Aziz and Rather [2].
3. Theorem and Comment on Theorem D
In this paper, firstly, we prove the following integral extension whose ordinary version corresponds to the corrected form of Theorem C which we state as the corollary. Secondly, we point out a flaw concerning Theorem D proved by Singh and Shah [13].
Theorem. Let
where
and
where
From (3.3) we have
This gives for
The above inequality holds for all points on
Let
Now, the reciprocal polynomial of
where
Applying Lemma 2.1 to the polynomial
where
Now,
Further, using Lemma 2.2 to the polynomial
Thus, using the fact of (3.7) to Lemma 2.3, we have
Now, for real numbers α and
which implies, for
For points
Then, for every
for points
Substituting the value of
Multiplying by
which is equivalent to
which completes the proof of the theorem.
Remark 3.1. If we let
where
Let
Choose the argument of λ suitably such that
Using (3.13) in (3.12), we have
Combining (3.11) and (3.14), we get
which on simplification gives
Now,
and
Using (3.16) and (3.17) in (3.15) and considering the limit as
Corollary Let
Remark 3.2. In fact, inequality (1.10) of Theorem C is not the correct form it should have been. It must be noted that in the correct form of (1.10), as is given by the corollary, every factor
Remark 3.3. In the theorem,
As
Comment on Theorem D. Theorem D is incorrect as it is based on an incorrect lemma (for reference see [5, Lemma 2.2]) as pointed out by Qazi [11].
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