### 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 ^{i𝜃})=0

Substituting the value of _{1}

Multiplying by

which is equivalent to

which completes the proof of the theorem.

**Remark 3.1.** If we let

where

Let _{0}

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 _{0}|

**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].

### References

- A. Aziz and Q. M. Dawood,
Inequalities for a polynomial and its derivatives , J. Approx. Theory,54 (1988), 306-313. - A. Aziz and N. A. Rather,
Some Zygmund type Lq inequalities for polynomials , J. Math. Anal. Appl.,289 (2004), 14-29. - S. Bernstein. Lecons sur les propriétés extrémales et la meilleure approximation desfonctions analytiques d'une variable réelle. Paris: Gauthier Villars; 1926.
- T. N. Chan and M. A. Malik,
On Erdös-Lax Theorem , Proc. Indian Acad. Sci. Math. Sci.,92(3) (1983), 191-193. - K. K. Dewan and S. Hans,
On maximum modulus for the derivative of a polynomial , Ann. Univ. Mariae Curie-Skodowska Sect. A,63 (2009), 55-62. - R. B. Gardner, N. K. Govil and S. R. Musukula,
Rate of growth of polynomials not vanishing inside a circle , J. Inequal. Pure Appl. Math.,6(2) (2005), 1-9. - P. D. Lax,
Proof of a conjecture of P. Erdös on the derivative of a polynomial , Bull. Amer. Math. Soc.,50 (1944), 509-513. - M. A. Malik,
On the derivative of a polynomial , J. London Math. Soc.,2(1) (1969), 57-60. - M. S. Pukhta, On extremal properties and location of zeros of polynomials, Ph.D Thesis, Jamia Millia Islamia, New Delhi(2002).
- M. A. Qazi,
On the maximum modulus of polynomials , Proc. Amer. Math. Soc.,115 (1992), 337-343. - M. A. Qazi,
Remarks on some recent results about polynomials with restricted zeros , Ann. Univ. Mariae Curie-Sklodowska Sect. A,67 (2013), 59-64. - W. Rudin. Real and Complex Analysis. Tata Mcgraw-Hill Publishing Company (Reprinted in India); 1997.
- G. Singh and W. M. Shah,
Some inequalities for derivatives of polynomials , Kyung-pook Math. J.,57 (2017), 537-543. - A. E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, Inc. New York(1958).