Abstract
In this paper, we consider the class of polynomials not vanishing in the unit disk and obtain a result that improves the results of Dubinin, Aziz and Dawood and the classical result of Ankeny and Rivlin.
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1 Introduction and statement of results
For an arbitrary entire function f(z), let \(M(f,r):=\max _{|z|=r}|f(z)|\). For a polynomial P(z) of degree n, it is known that
Inequality (1.1) is a simple consequence of Maximum Modulus Principle (see [4]). It was shown by Ankeny and Rivlin [1] that if \(P(z)\ne 0\) in \(|z|< 1\), then (1.1) can be replaced by
In 1988, Aziz and Dawood further improved the bound in (1.2) and proved under the same hypothesis that
Recently, Dubinin [3] obtained the following refinement of (1.2) by using the classical Schwarz lemma.
Theorem 1
If\(P(z)=\sum \nolimits _{j=0}^{n}c_jz^j\)is a polynomial of degree\(n\ge 2\)with no zeros in\(|z|< 1\), then for any\(\rho >1\),
The result is best possible and equality holds in (1.4) for \(P(z)=\frac{\mu +\nu z^n}{2}\), \(|\mu |=|\nu |=1.\)
In this note, we prove the following generalization of (1.4) which sharpens the bounds in (1.2) and (1.3) as well.
Theorem 2
If\(P(z)=\sum \nolimits _{j=0}^{n}c_jz^j\)is a polynomial of degree\(n\ge 2\)with no zeros in\(|z|< 1\), then for any\(\rho >1\)and\(0\le t\le 1,\)
where \(m=min_{|z|=1}|P(z)|.\)
The result is best possible and equality holds in (1.5) for \(P(z)=\frac{\mu +\nu z^n}{2}\), \(|\mu |=|\nu |=1.\)
Remark 1
Since if \(P(z)=\sum \nolimits _{j=0}^{n}c_jz^j\ne 0\) in \(|z|< 1,\) then \(|c_0|\ge |c_n|.\) Also, as in the proof of the Theorem 2 (given in the next section), we have for every \(\lambda \) with \(|\lambda |\le 1,\) the polynomial \(P(z)-\lambda m\) does not vanish in \(|z|<1,\) hence
If in (1.6), we choose the argument of \(\lambda \) suitably and note that \(|c_0|>m\) (from (2.2), proof of Theorem 2), we get
If we take \(|\lambda |=t\) in (1.7) so that \(0\le t\le 1\), we get \(tm+|c_n|\le |c_0|.\)
Remark 2
Here, we show that for \(\rho >1,\)
which is equivalent to showing
that is
which clearly holds by Remark 1. Also, the function \(xM(P,1)-(x-1)tm\) is a non-decreasing function of x. If we combine this fact with (1.8) according to which
it follows that the right hand side of (1.5) does not exceed \((\frac{1+\rho ^n}{2})M(P,1)-(\frac{\rho ^n-1}{2}) t m,\) we have a refinement of (1.3).
Remark 3
2 Proof of Theorem
Proof of Theorem 2. Since \(P(z)=\sum \nolimits _{j=0}^{n}c_jz^j\) has all its zeros in \(|z|\ge 1\) and \(m=\min _{|z|=1}|P(z)|\), therefore
It follows by the Maximum and Minimum Modulus Principles that the strict inequality
holds for \(|z|<1\).
We show that for every complex \(\alpha \) with \(|\alpha |\le 1\), the polynomial \(F(z)=P(z)-\alpha m\) does not vanish in \(|z|< 1\). For if \(F(z)=P(z)-\alpha m\) has a zero in \(|z|<1,\) say at \(z=z_1\) with \(|z_1|<1,\) then
This gives,
where \(|z_{1}|< 1\), which contradicts (2.2).
Hence, we conclude that the polynomial F(z) does not vanish in \(|z|< 1\). Applying Theorem 1 to the polynomial \(F(z)=P(z)-\alpha m=(c_{0}-\alpha m)+\sum \nolimits _{j=1}^{n}c_{j}z^{j}\), we get for every complex \(\alpha \) with \(|\alpha |\le 1\) and \(\rho > 1\),
For every \(\alpha \) with \(|\alpha |\le 1,\) we have
since \(|\alpha |m \le m < |P(0)| = |c_0|,\) by (2.2).
Further, the function \(\big (\frac{x+\rho |c_n|}{x+|c_n|}\big )\) is decreasing on \(\{x:x>-|c_n|\}\cup \{x:x<-|c_n|\}\) for every \(\rho >1,\) it follows from (2.3) that for every \(\alpha \) with \(|\alpha |\le 1\) and for every \(\rho >1,\)
where \(z_0\) is a point on \(|z|=1\) such that \(|P(z_0)|=M(P,1).\) Also by (2.1) and (2.2), we have
we take in particular \(z=z_0\) in (2.5) and get
Choosing the argument of \(\alpha \) with \(|\alpha |\le 1\) on the right hand side of (2.4) such that
which is possible by (2.6), we obtain from (2.4) that
for every \(\alpha \) with \(|\alpha |\le 1\) and for every \(\rho >1\).
The above inequality is equivalent to (1.5) and this completes the proof of Theorem 2.
References
Ankeny, N.C., and T.J. Rivlin. 1955. On a theorem of S. Bernstein. Pacific Journal of Mathematics 5: 849–852.
Aziz, A., and Q.M. Dawood. 1988. Inequalities for a polynomial and its derivative. Journal of Approximation Theory 54: 306–313.
Dubinin, V.N. 2007. Applications of the Schwarz lemma to inequalities for entire functions with constarints on zeros. Journal of Mathematical Sciences 143: 3069–3076.
Milovanovic, G.V., D.S. Mitrinovic, and Th. M Rassias. 1944. Topics in polynomials. Extremal problems, Inequalities, Zeros. Singapore: World Scientific.
Acknowledgements
The authors are highly grateful to the anonymous referee for the valuable suggestions regarding the paper.
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Mir, A., Hussain, I. & Wani, A. A note on Ankeny–Rivlin theorem. J Anal 27, 1103–1107 (2019). https://doi.org/10.1007/s41478-018-0091-8
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DOI: https://doi.org/10.1007/s41478-018-0091-8