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
Received: December 6, 2020; Revised: October 12, 2021; Accepted: November 8, 2021
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
Thus, it is appropriate to denote
Further, if we define
Equality holds in (1.6) for
Theorem A. Let
Equality occurs in (1.8) for
As an improvement of Theorem A, Pukhta  proved
Theorem B. Let
Equality in (1.9) occurs for
Singh and Shah  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 , the authors further proved the following result as generalization of Theorem C.
Theorem D. Let
We need the following lemmas to prove the theorem.
Lemma 2.1. If
This lemma is due to Qazi .
This lemma is due to Gardner et al. .
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 .
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 .
From (3.3) we have
This gives for
The above inequality holds for all points on
Now, the reciprocal polynomial of
Applying Lemma 2.1 to the polynomial
Further, using Lemma 2.2 to the polynomial
Now, for real numbers α and
which implies, for
Then, for every
Substituting the value of
which is equivalent to
which completes the proof of the theorem.
Remark 3.1. If we let
Choose the argument of λ suitably such that
which on simplification gives
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
- 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).