Abstract
Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number \(\alpha \) is defined by
If \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j}\), \(0\le s\le n,\) has all its zeros in \(|z| \le k, k\ge 1\), then for \( |\alpha |\ge k\), Kumar and Dhankhar [Bull, Math. Soc. Sci. Math., 63(4), 359-367 (2020)] proved
In this paper, we first improve the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta. Math. Hungar., 104, 115-126 (2004)] and also prove an inequality for a subclass of polynomials having no zero in \( |z |< k, k\le 1\).
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1 Introduction
Let \(p(z)=\displaystyle {\sum _{j=0}^{n}c_jz^j}\) be a polynomial of degree n over the set of complex numbers. We will use q(z) to represent the polynomial \(z^n\overline{p\left( \frac{1}{\bar{z}}\right) }\).
According to the famous Bernstein’s inequality [6],
Equality in (1) holds for \(p(z)=\alpha z^n, \alpha \ne 0\).
If we restrict the zeros of p(z), inequality (1) can be refined. In this direction, Erdös conjectured and later Lax [19, p. 1] proved that if p(z) is a polynomial of degree n having no zero in \( |z |< 1\), then
Inequality (2) is best possible for \(p(z)=\alpha +\beta z^n\), where \( |\alpha |= |\beta |\).
It was R. P. Boas who asked that if p(z) is a polynomial of degree n not vanishing in \( |z |<k\), \(k>0\), then how large
A partial answer to this problem was given by Malik [20, Theorem, p. 58], who proved that if p(z) is a polynomial of degree n having no zeros in \( |z |<k\), \(k\ge 1\), then
In the literature, there exist generalizations and improvements of inequality (3), for brief understanding one can refer to: Chan and Malik [8], Qazi [21], Bidkham and Dewan [7], Aziz and Zargar [4], Chanam and Dewan [9], Aziz and Shah [3] etc.
On the other hand, for the class of polynomials p(z) such that \(p(z)\ne 0\) for \( |z |<k\), \(k\le 1\), the precise estimate for maximum of \( |p'(z) |\) on \( |z |=1\) does not seem to be easily obtainable. For quit some time, it was believed that the inequality analogous to (3) for \(p(z)\ne 0\) in \( |z |<k\), \(k\le 1\), should be
till E. B. Saff gave the example \(p(z)=\left( z-\frac{1}{2}\right) \left( z+\frac{1}{3}\right) \) to counter this belief.
With extra assumption inequality (4) could be satisfied. In this direction, Govil [11] proved that if p(z) is a polynomial of degree n having no zero in \( |z |<k\), \(k\le 1\), with additional hypothesis that \( |p'(z) |\) and \( |q'(z) |\) attain their maxima at the same point on \( |z |=1\), then
Under the same set of hypothesis, Kumar and Dhankar [18, Theorem 2] further improved inequality (5) by proving
Another improvement of (5) was also recently obtained by Singh and Chanam [23, Theorem 3] by proving
In 1939, Turán [26] provided a lower bound estimate of the derivative to the size of the polynomial by restricting its zeros, and proved that if p(z) has all its zeros in \( |z |\le 1\), then
Aziz and Dawood [1, Theorem 4] further refined inequality (8) by involving \(\displaystyle {\min _{ |z |=1} |p(z) |}\). In fact, they proved
Both the inequalities (8) and (9) are best possible and equality holds if p(z) has all its zeros on \( |z |=1\).
Inequalities (8) and (9) have been extended and generalized in different directions (see [3, 5, 12,13,14]). For polynomial p(z) having all its zeros in \( |z |\le k, k\ge 1,\) Govil [12, Theorem, p. 544] proved that
Further, as an improvement of (10) and a generalization of (9), Govil [13, Theorem 2] proved
Inequalities (10) and (11) are sharp and equality holds for \(p(z)=z^n+k^n\).
The concept of ordinary derivative of a polynomial has been generalized to polar derivative of a polynomial as follows:
If p(z) is a polynomial of degree n and \(\alpha \) be any real or complex number, the polar derivative of p(z) with respect to \(\alpha \), denoted by \(D_{\alpha }p(z)\), is defined as
It is easy to see that \(D_{\alpha }p(z)\) is a polynomial of degree at most \(n-1\) and it generalizes the ordinary derivative in the sense that
Shah [22] extended inequality (8) to the polar derivative and proved that if p(z) is a polynomial of degree n having all its zeros in \( |z |\le 1,\) then for any complex number \(\alpha \) with \( |\alpha |\ge 1\)
Recently, Gulzar et al. [17, Theorem 2.1] refined inequality (12) and proved that if \(p(z)=\displaystyle {\sum _{j=0}^{n}c_jz^j}\) is a polynomial of degree n having all its zeros in \( |z |\le 1,\) then for any complex number \(\alpha \) with \( |\alpha |\ge 1\) and \( |z |=1\)
In 1998, Aziz and Rather [2, Theorem 2] extended inequality (10) to polar derivative by proving that if p(z) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1\), then for every complex number \(\alpha \) with \( |\alpha |\ge k\),
Recently, Kumar and Dhankhar [18, Theorem 3] obtained a generalization as well as improvement of (14) by establishing that if \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j},0\le s\le n,\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1,\) then for any complex number \(\alpha \) with \( |\alpha |\ge k,\)
With the same hypothesis, Singh and Chanam [23, Theorem 1] provided another improvement of (14) and a generalization of (13) and obtained
Govil and Mctume [15, Theorem 3] extended inequality (11) to polar derivative and proved
where \(\alpha \) is any complex number with \( |\alpha |\ge 1+k+k^n\).
Improvements of inequality (17) by involving leading coefficient and constant term of the polynomial can be seen in recent works of Singh and Chanam [23, Theorem 2] and Singh et al. [24, Theorem 4].
2 Main results
We begin by presenting the following refinement of inequality (15) and inequality (16).
Theorem 1
If \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j},0\le s\le n,\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1,\) then for any complex number \(\alpha \) with \( |\alpha |\ge k\),
Remark 1
Since the polynomial \(h(z)=\frac{p(z)}{z^s}=\displaystyle {\sum _{j=0}^{n-s}c_jz^j}\) has all its zeros in \( |z |\le k, k\ge 1\), we have
which is equivalent to
and
Dividing both sides of (18) by \( |\alpha |\) and taking limit as \( |\alpha |\rightarrow \infty \), we get the following generalization and refinement of inequality (10) due to Govil [12].
Corollary 1
If \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j},0\le s\le n,\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1,\) then
When \(s=0\), Theorem 1, in particular, gives the following improvement of inequality (14) proved by Aziz and Rather [2] and a generalization and an improvement of inequality (13) of Gulzar et al. [17].
Corollary 2
If \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1\), then for any complex number \( |\alpha |\) with \( |\alpha |\ge k\)
Dividing both sides of (20) by \( |\alpha |\) and taking limit as \( |\alpha |\rightarrow \infty \), we get the following refinement of inequality (10) due to Govil [12].
Corollary 3
If \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1\), then
The inequality (21) is best possible for \(p(z)=z^n+k^n\).
Remark 2
Taking \(k=1\) in Corollary 3, inequality (21) provides a refinement of inequality (8) due to Turán [26].
As an application of Theorem 1, we obtain the following result which is a refinement of inequality (17) due to Govil and Mctume [15] and a result recently proved by Singh and Chanam [23, Theorem 2].
Theorem 2
If \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1\), then for any complex number \(\alpha \) with \( |\alpha |\ge 1+k+k^n\)
where \(m=\displaystyle {\min _{ |z |=k} |p(z) |}\) and \(\theta _{0}=\arg \left\{ p(e^{i\phi _{0}})\right\} \) such that \( |p(e^{i\phi _{0}}) |=\displaystyle {\max _{ |z |=1} |p(z) |}\).
Remark 3
If \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1,\) then for any complex number \( |\lambda |e^{i\theta _{0}}\) with \( |\lambda |<1\), by Rouche’s theorem it follows that the polynomial \(p(z)+ |\lambda |e^{i\theta _{0}} m=(c_0+ |\lambda |e^{i\theta _{0}} m)+c_1z+\cdots +c_nz^n\) has all its zeros in \( |z |\le k\), where \(m=\displaystyle {\min _{ |z |=k} |p(z) |}\), then
which implies that
Taking \( |\lambda |\rightarrow 1\), we get
and
Remark 4
Dividing both sides of (22) by \( |\alpha |\) and taking limit as \( |\alpha |\rightarrow \infty \), we have the following refinement of inequality (11) due to Govil [13].
Corollary 4
If \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) is a polynomial of degree n having all its zeros in \( |z |\le k, k\ge 1\), then
where \(m=\displaystyle {\min _{ |z |=k}p(z)}\) and \(\theta _{0}=\arg \left\{ p(e^{i\phi _{0}})\right\} \) such that \( |p(e^{i\phi _{0}}) |=\displaystyle {\max _{ |z |=1} |p(z) |}\).
Inequality (23) is best possible for \(p(z)=z^n+k^n\).
Remark 5
Taking \(k=1\) in Corollary 4, inequality (23) reduces to a refinement of inequality (9) due to Aziz and Dawood [1].
Corollary 5
If \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) is a polynomial of degree n having all its zeros in \( |z |\le 1\), then
where \(m=\displaystyle {\min _{ |z |=1}} |p(z) |\) and \(\theta _{0}=\arg \left\{ p(e^{i\phi _{0}})\right\} \) such that \( |p(e^{i\phi _{0}}) |=\displaystyle {\max _{ |z |=1} |p(z) |}\).
Further, we are able to prove an improvement of inequalities (6) and (7).
Theorem 3
If \(p(z)=\displaystyle {\sum _{j=0}^{n}}c_jz^j\) is a polynomial of degree n having no zero in \( |z |< k, k\le 1\). If \( |p'(z) |\) and \( |q'(z) |\) attain their maxima at the same point on \( |z |=1\), then
The result is sharp and equality in (25) holds for \(p(z)=z^n+k^n\).
Remark 6
Since \(p(z)=\displaystyle \sum _{j=0}^{n}c_jz^j\) has all its zeros in \( |z |\ge k,k\le 1\), q(z) has all its zeros in \( |z |\le \frac{1}{k}, \frac{1}{k}\ge 1\), then
which equivalently gives
and
From inequalities (26) and (27), it is evident that the bound (25) improves both the bounds given by (6) and (7).
Remark 7
Taking \(k=1\) in Theorem 3, we get the following improvement of (2) due to Erdös and Lax for a subclass of polynomials.
Corollary 6
If \(p(z)=\displaystyle {\sum _{j=0}^{n}}c_jz^j\) is a polynomial of degree n having no zero in \( |z |< 1\). If \( |p'(z) |\) and \( |q'(z) |\) attain their maxima at the same point on \( |z |=1\), then
3 Lemmas
We need the following lemmas to prove our theorems.
Lemma 1
If \(p(z)=\displaystyle {\sum _{j=0}^{n}c_jz^j}\) is a polynomial of degree \(n\ge 1\) having all its zeros in \( |z |\le 1,\) then for all z on \( |z |=1\) with \(p(z)\ne 0\).
The above result is due to Dubin [10, Theorem 4]( also see Singh and Chanam [23, Lemma 3] and Wali and Shah [25, Inequality 9]).
Lemma 2
Let \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j},0\le s\le n\) be a polynomial of degree n having all its zeros in \( |z |\le k,k\ge 1,\) then
The above result appears in Kumar and Dhankar [18, Lemma 4].
Lemma 3
If \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j},0\le s\le n\) is a polynomial of degree n having all its zeros in \( |z |\le 1,\) with \(s-\)fold zeros at the origin, then for any complex number \(\alpha \) with \( |\alpha |\ge 1\) and on \( |z |=1\)
This result appears in Singh and Chanam [23, Lemma 5].
Lemma 4
If p(z) is a polynomial of degree n, then on \( |z |=1\)
The above result is a particular case of a result [16, Inequality 3.2] due to Govil and Rahman.
4 Proofs of the theorems
Proof of Theorem 1
Since \(p(z)=z^s\displaystyle {\sum _{j=0}^{n-s}c_jz^j}\) has all its zeros in \( |z |\le k,k\ge 1,\) the polynomial \(p(kz)=z^s\left( k^sc_0+k^{s+1}c_1z+\cdots k^{n}c_nz^{n-s}\right) \) has all its zeros in \( |z |\le 1\). Using Lemma 3 to p(kz), we get for \( |\frac{\alpha }{k} |\ge 1\)
that is
Using Lemma 2 and the fact that \(\displaystyle {\max _{ |z |=1}}|np(kz)+\left( \frac{\alpha }{k}-z\right) kp'(kz)|=\displaystyle {\max _{ |z |=k} |D_{\alpha }p(z) |}\), inequality (33) implies
As we can see that \(D_{\alpha }p(z)\) is a polynomial of degree at most \(n-1\) and \(k\ge 1\), it is well-known that
\(\displaystyle {\max _{ |z |=k} |D_{\alpha }p(z) |\le k^{n-1}\max _{ |z |=1} |D_{\alpha }p(z) |}\). Using this fact, inequality (34) gives
which gives inequality (18), and the proof of Theorem 1 is complete. \(\square \)
Proof of Theorem 2
If p(z) has a zero on \( |z |=k,\) then \(m=0\) and the result follows trivially from Theorem 1. So, without loss of generality, let us assume that p(z) has all its zeros in \( |z |<k,k\ge 1,\) then it follows by Rouche’s theorem that for any complex number \(\lambda \) with \( |\lambda |<1\), the polynomial \(p(z)+\lambda m=(c_{0}+\lambda m)+c_1z+\cdots +c_nz^n\) has all its zeros in \( |z |<k,k\ge 1\). Therefore, applying Theorem 1 to \(p(z)+\lambda m\) with \(s=0\), we get for \( |\alpha |\ge 1+k+k^n\)
Let \(0\le \phi _0<2\pi \), be such that \(|p(e^{i\phi _0})|=\displaystyle {\max _{ |z |=1} |p(z) |}\). Then, inequality (35) takes
Now,
Setting the argument \(\phi \) such that \(\phi =\theta _{0}\), then
Using this fact in inequality (37), we have
which is equivalent to
Taking \( |\lambda |\rightarrow 1\), the above inequality reduces to (22). This completes the proof of Theorem 2. \(\square \)
Proof of Theorem 3
Since p(z) has all its zeros in \( |z |\ge k,k\le 1\), q(z) has all its zeros in \( |z |\le \frac{1}{k}, \frac{1}{k}\ge 1\). Then applying Corollary 3 to q(z), we have
By Lemma 4, we have on \( |z |=1\),
Since \( |p'(z) |\) and \( |q'(z) |\) attain their maxima at the same point on \( |z |=1\), then
Combining (39), (40) and (41), we have
which is equivalent to
\(\square \)
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Singh, T.B., Soraisam, R. & Chanam, B. Sharpening of Turán type inequalities for polar derivative of a polynomial. Complex Anal Synerg 9, 3 (2023). https://doi.org/10.1007/s40627-023-00113-x
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DOI: https://doi.org/10.1007/s40627-023-00113-x