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# On Bernstein and Turán-type integral mean estimates for polar derivative of a polynomial

*Journal of Inequalities and Applications*
**volume 2024**, Article number: 107 (2024)

## Abstract

Let \(p(z)\) be a polynomial of degree *n* having no zero in \(|z|< k\), \(k\leq 1\), then Govil [*Proc. Nat. Acad. Sci.,* 50(1980), 50-52] proved

provided \(|p'(z)|\) and \(|q'(z)|\) attain their maxima at the same point on the circle \(|z|=1\), where

In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results.

## 1 Introduction

Empirical findings and inquiries across diverse realms of science and engineering frequently undergo a transformation into mathematical representations and mathematical frameworks. Nearly every realm within mathematics, ranging from algebraic number theory and algebraic geometry to applied analysis, Fourier analysis, numerical analysis, and computer science, possesses its own collection of principles and concepts stemming from the examination of polynomial functions.

Historically, the question relating to polynomials, for example, the solution of polynomial equations and the approximation by polynomials, gives rise to some of the most important problems of the day. The well-known Russian mathematician Chebyshev (1821–1894) studied some properties of polynomials with the least deviation from a given continuous function and introduced the concept of best approximation in mathematical analysis. Various interesting inequalities concerning the estimate of the sup-norm of the derivative as an upper bound in terms of the sup-norm of the polynomial itself, known as Bernstein-type inequalities, play a key role in the literature for proving the inverse theorem in approximation theory (see Borwein and Erdélyi [8], Ivanov [28], Lorentz [31], Telyakovskii [47]) and, of course, have their own intrinsic interests. The first result in this area was associated with the research of the famous Russian chemist Mendeleev [36]. In fact, Mendeleev’s problem was to determine \(\displaystyle{\max _{-1\leq x\leq 1}|p'(x)|}\), where \(p(x)\) is a quadratic polynomial of real variable *x* with real coefficients and satisfying \(-1\leq p(x)\leq 1\) for \(-1\leq x\leq 1\). He was only able to prove that if \(p(x)\) is a quadratic polynomial and \(|p(x)|\leq 1\) on \([-1,1]\), then \(|p'(x)|\leq 4\) on the same interval. Markov [35] generalized this result for a polynomial of degree *n* in the real line. In fact, he proved that if \(p(x)\) is an algebraic polynomial of degree at most *n* with real coefficients, then

After about twenty years, Bernstein [6] needed the analog of Markov’s theorem for the unit disc in the complex plane instead of the interval \([-1,1]\) in order to prove inverse theorem in approximation (see Borwein and Erdélyi [8, p. 241]) to estimate how well a polynomial of a certain degree approximates a given continuous function in terms of its derivatives and Lipschitz constants. This leads to the famous well-known result known as Bernstein’s inequality that states that if \(t\in \tau _{n}\) (the set of all real trigonometric polynomials of degree at most *n*), then for \(K:=[0,2\pi )\),

The above inequality remains true for all \(t\in \tau _{n}^{c}\) (the set of all complex trigonometric polynomials of degree at most *n*), which implies, as a particular case, the following algebraic polynomial version of Bernstein’s inequality on the unit disk.

### Theorem 1.1

*If* \(p(z)\) *is a polynomial of degree* *n*, *then*

Equality holds in (2) if \(p(z)\) has all its zeros at the origin.

It is really of interest both in theoretical and practical aspects that continuous functions are approximated by polynomials. One such approach of approximation is made through the applications of Bernstein’s inequality, particularly the trigonometric version, and, in this regard, we have the following interesting result (Theorem 1.2) [8, p. 241, Part (a) of E.18], which approximates *m* times differentiable real-valued function on a half-closed interval \([0,2\pi )\) by trigonometric polynomials. For the sake of convenience of the readers, we state the above result more precisely.

Let Lip_{α}, \(\alpha \in (0,1]\), denote the family of all real-valued functions *g* defined on *K* satisfying

If \(C(K)\) denotes the set of all continuous functions on *K*, then for \(f\in C(K)\), let

### Theorem 1.2

(*Direct theorem*) *Suppose* *f* *is* *m* *times differentiable on* *K* *and* \(f^{(m)}\in \mathrm{Lip}_{\alpha}\) *for some* \(\alpha \in (0,1]\). *Then*, *there is a constant* *C* *depending only on* *f* *so that*

On the other hand, the converse (inverse) of Theorem 1.2 is essentially of interest and is stated below.

### Theorem 1.3

(*Inverse theorem*) *Suppose* *m* *is a nonnegative integer*, \(\alpha \in (0,1)\), *and* \(f\in C(K)\). *Suppose there is a constant* \(C>0\) *depending only on* *f* *such that*

*Then* *f* *is* *m* *times continuously differentiable on* *K* *and* \(f^{(m)}\in \mathrm{Lip}_{\alpha}\).

The proof of Theorem 1.3 is done by the application of the well-known result due to Bernstein (inequality (1)) given in [8].

From the above discussion, it is worth noting that Bernstein- and Markov-type inequalities play a significant role in approximation theory. Direct and inverse theorems of approximation and related matters may be found in many books on approximation theory, including Cheney [15], Lorentz [31], and DeVore and Lorentz [18].

Moreover, inequality (2) shows how fast a polynomial of degree at most *n* can change, and is of interest both in mathematics, especially in approximation theory, and in the application areas such as physical systems. Various analogs of this inequality are known in which the underlying intervals, the sup-norms, and the families of polynomials, are replaced by more general sets, norms, and families of functions, respectively. One such generalization is replacing sup-norm by a factor involving the integral mean.

Let \(p(z)\) be a polynomial of degree *n* over the set of complex numbers and \(q(z)=z^{n}\overline{p(\frac{1}{\overline{z}})}\). We define the integral mean of \(p(z)\) on the unit circle \(|z|=1\) by

If we let \(r\rightarrow \infty \) in (3) and make use of the well-known fact from analysis (see [43, 46]) that

we can suitably denote

Similarly, we can define

and show that \(\lim \limits _{r\rightarrow 0^{+}}\|p\|_{r}=\|p\|_{0}\). It would be of further interest that by taking the limit as \(\lim \limits _{r\rightarrow 0^{+}}\), the stated results, holding for \(r>0\), also hold for \(r=0\). Inequality (2) can be obtained by letting \(r\rightarrow \infty \) in the inequality

Inequality (6) was proved by Zygmund [49] for \(r\geq 1\) and by Arestov [1] for \(0< r<1\).

If we restrict to the class of polynomials having no zero in \(|z|<1\), then inequalities (2) and (6) can be respectively improved as

Inequality (7) was conjectured by Erdös and later verified by Lax [30] whereas inequality (8) was proved by de Bruijn [16] for \(r\geq 1\) and by Rahman and Schmeisser [39] for \(0< r<1\). On the other hand, in 1939 (see [48]), Turán obtained a lower bound for the maximum of \(|p'(z)|\) on \(|z|=1\) by proving that if \(p(z)\) is a polynomial of degree *n* having all its zeros in \(|z|\leq 1\), then

As a generalization of (9), Govil [24] proved that if \(p(z)\) is a polynomial of degree *n* having all its zeros in \(|z|\leq k\), \(k\geq 1\), then

As a generalization of (7), Malik [32] proved that if \(p(z)\) is a polynomial of degree *n* having no zeros in \(|z|< k\), \(k\geq 1\), then

Under the same hypotheses of the polynomial \(p(z)\), Govil and Rahman [26] extended inequality (11) to \(L^{r}\) norm by showing that

Gardner and Weems [23] and independently Rather [41] showed that inequality (12) also holds true for \(0< r<1\). There are many extensions of inequality (12) (see, Chan and Malik [13], Dewan and Bidkham [19], Chanam and Dewan [14] and Dewan and Mir [20]).

For the class of polynomials not vanishing in \(|z|< k\), \(k\leq 1\), the precise upper bound estimate for the maximum of \(|p'(z)|\) on \(|z|=1\), in general, does not seem to be easily obtainable. For quite some time, it was believed that if \(p(z)\) has no zero in \(|z|< k\), \(k\leq 1\), then the inequality that generalizes (7) should be

until E. B. Saff gave the example \(p(z)=\left (z-\frac{1}{2}\right )\left (z+\frac{1}{3}\right )\) to counter this belief.

Thus, the approximation does not seem to be known in general, and this problem is still open. However, some special cases in this direction have been considered by many people where some partial extensions of (7) are established. In 1980, it was again Govil [25], who generalized (7) with an extra condition by proving that

### Theorem 1.4

*If* \(p(z)\) *is a polynomial of degree* *n* *having no zeros in* \(|z|< k\), \(k\leq 1\), *then*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\), *where throughout the paper*

For the first time in 1984, Malik [33] extended inequality (9) proved by Turán [48] into integral mean version and proved that if \(p(z)\) is a polynomial of degree *n* having all its zeros in \(|z|\leq 1\), then for \(r>0\),

The result is sharp, and the equality holds for \(p(z)=(z+1)^{n}\).

In 1988, Aziz [2] obtained the integral mean extension of inequality (10) and proved the following theorem.

### Theorem 1.5

*If* \(p(z)\) *is a polynomial of degree* *n* *having all its zeros in* \(|z|\leq k\), \(k\geq 1\), *then for* \(r\geq 1\),

On the other hand, in 2021, Devi et al. [17] proved the following more general result, which gives inequality (13) as a special case.

### Theorem 1.6

*If* \(p(z)\) *is a polynomial of degree* *n* *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every* \(r>0\),

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

Devi et al. [17] also proved another result that sharpened Theorem 1.6.

### Theorem 1.7

*If* \(p(z)\) *is a polynomial of degree* *n* *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every real or complex number* *α* *with* \(|\alpha |<1\) *and for every* \(r>0\),

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\), *where throughout the paper* \(m=\min \limits _{|z|=k}|p(z)|\).

Before proceeding to some other results, let us introduce the concept of the polar derivative involved. For a polynomial \(p(z)\) of degree *n*, we define

the polar derivative of \(p(z)\) with respect to the point *β* (see [34] and [22, Chap. 6]). The polynomial \(D_{\beta}{p(z)}\) is of degree at most \(n-1\), and it generalizes the ordinary derivative \(p'(z)\) in the sense that

uniformly with respect to *z* for \(|z|\leq R\), \(R>0\).

Various results concerning the polar derivative of a polynomial can be found in the comprehensive books by Milovanović et al. [37], Marden [34], and Rahman and Schmeisser [40], where some approaches to obtaining polynomial inequalities are developed on applying the methods and results of the geometric function theory.

In 1998, Aziz and Rather [5] established the polar derivative generalization of (10) by proving that if \(p(z)\) is a polynomial of degree *n* having all its zeros in \(|z|\leq k\), \(k\geq 1\), then for every complex number *β* with \(|\beta |\geq k\),

whereas, the polar derivative analog of (13) was recently given by Mir and Breaz [38]. They proved that if \(p(z)\) is a polynomial of degree *n*, which does not vanish in \(|z|< k\), \(k\leq 1\), then for every complex number *β* with \(|\beta |\geq 1\),

provided \(|p'(z)|\) and \(|q'(z)|\) attain their maxima at the same point on \(|z|=1\).

They [38] also proved that if \(p(z)\) is a polynomial of degree *n*, which does not vanish in \(|z|< k\), \(k\leq 1\), then for every complex number *β* with \(|\beta |\geq 1\),

provided \(|p'(z)|\) and \(|q'(z)|\) attain their maxima at the same point on \(|z|=1\).

In an attempt to obtain the integral version of inequality (19) due to Aziz and Rather [5], it was only in 2017 that Rather and Bhatt [42] obtained the integral setting of this inequality. Very recently, in 2023, Singha and Chanam [45] proved a result that is an integral extension of a result due to Kumar [29] in this direction. In this paper, our first main goal is to establish an integral version of a refinement of inequality (19) due to Aziz and Rather [5], which further improves the well-known inequality (9) due to Túran [48] for ordinary derivative. As applications of one of the interesting consequences of the above result, further, we have been able to obtain improved versions of both the inequalities recently proved by Mir and Breaz [38] concerning polar derivatives. Our paper is organized as follows. In Sect. 2, we state and discuss the interesting consequences of our main results. In Sect. 3, we present some auxiliary results necessary to prove our main results. The proof of our main results will be postponed to Sect. 4 of our paper.

## 2 Main results

This section is devoted to stating our main observations as theorems and corollaries. The importance of the first theorem is profound in that we can have a rich number of key consequences to some of the already obtained polynomial inequalities in this direction; for instance, we first deduce a result from which an interesting corollary (e.g., Corollary 2.5) is also obtained, and as some chief applications of this corollary, we further prove two theorems from which various interesting implications to the existing known results follow.

### Theorem 2.1

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having all its zeros in* \(|z|\leq k\), \(k\geq 1\), *then for every complex number* *α* *and* *β* *with* \(|\alpha |<1\), \(|\beta |\geq k\) *and for each* \(r>0\),

*and*

### Remark 2.2

For polynomial of degree 1, simply \(p(z)=a_{0}+a_{1}z\) and hence, we can easily evaluate \(\displaystyle{m=\min _{|z|=k}|p(z)|}=k|a_{1}|-|a_{0}|\) and \(\displaystyle{\max _{|z|=1}|D_{\beta}p(z)+\alpha m|}=\left |a_{0}+ \beta a_{1}+\alpha (k|a_{1}|-|a_{0}|)\right |\). So, we need not find their estimates as their exact values are readily obtainable.

### Remark 2.3

If \(\alpha =0\), inequalities (22) and (23) of Theorem 2.1 amount to a remarkable result, which refines a result due to Rather and Bhat [42, Theorem 1].

### Remark 2.4

Dividing both inequalities (22) and (23) of Theorem 2.1 by \(|\beta |\) and allowing \(|\beta |\rightarrow \infty \) yield

and

Putting \(\alpha =0\) in inequalities (24) and (25), the following result follows improving inequality (16) due to Aziz [2] and gives a refinement and generalization of inequality (15) due to Malik [33] for the polynomial with degree \(n\geq 2\).

### Corollary 2.5

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having all its zeros in* \(|z|\leq k\), \(k\geq 1\), *then for every* \(r>0\),

*and*

### Remark 2.6

Taking limit as \(r\rightarrow \infty \) on both sides of (24) and (25) yields

and

Let \(z_{0}\) on \(|z|=1\) be such that \(\displaystyle{\max _{|z|=1}|p(z)|=|p(z_{0})|}\). Then, in particular,

We choose a suitable argument for *α* such that

Applying (31) to (30), we have

Applying (32) to inequalities (28) and (29), we have

and

which is equivalent to

and

which vividly is a refinement and a generalization of a result due to Aziz and Dawood [4, Theorem 4] for \(n\geq 2\).

### Remark 2.7

If \(\alpha =0\), inequalities (33) and (34) amount to the following interesting consequence that is a refinement of inequality (10) due to Govil [24], and this improves as well as generalizes inequality (9) due to Tuŕan [48] for \(n\geq 2\).

### Corollary 2.8

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having all its zeros in* \(|z|\leq k\), \(k\geq 1\), *then*

*and*

### Remark 2.9

Again, letting \(r\rightarrow \infty \) on both sides of inequalities (22) and (23) of Theorem 2.1 and following similar arguments of Remark 2.6 for routing inequality (32) namely

\(\displaystyle{\max _{|z|=1}|p(z)+\alpha m|\geq \max _{|z|=1}|p(z)|+| \alpha |m}\) for some suitable *α*, we have

and

Considering \(\alpha =0\) in the above two inequalities, we get an interesting result that is a refinement of inequality (19) due to Aziz and Rather [5] for \(n\geq 2\).

### Corollary 2.10

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having all its zeros in* \(|z|\leq k\), \(k\geq 1\), *then for every complex number* *β* *with* \(|\beta |\geq k\),

*and*

As a first application of Corollary 2.5, we also establish the following improved integral setting of inequality (20) concerning polar derivative, and, as a particular case, our result also yields an improvement of Theorem 1.6 due to Devi et al. [17]. In fact, we prove

### Theorem 2.11

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every* \(r>0\) *and for every complex number* *β* *with* \(|\beta |\geq 1\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.12

As in Remark 2.2, for the class of polynomials having no zeros in \(|z|< k\), \(k\leq 1\), for polynomial of degree 1, simply \(p(z)=a_{0}+a_{1}z\), and hence, we can easily evaluate \(\displaystyle{\max _{|z|=1}|p(z)|}=|a_{0}|+|a_{1}|\), \(\displaystyle{m=\min _{|z|=k}|p(z)|}=|a_{0}|-k|a_{1}|\) and \(\displaystyle{\max _{|z|=1}|D_{\beta}p(z)|}=|a_{0}+\beta a_{1}|\). So, we need not find their estimates as their exact values are readily obtainable.

### Remark 2.13

Dividing inequalities (39) and (40) of Theorem 2.11 by \(|\beta |\) and letting \(|\beta |\rightarrow \infty \), we get the following interesting result that sharpens the bound of Theorem 1.6 due to Devi et al. [17] for the polynomial with degree \(n\geq 2\).

### Corollary 2.14

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every* \(r>0\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

Further, we prove another result involving \(m=\min \limits _{|z|=k}|p(z)|\), which is an integral extension of a more refined result of inequality (21) due to Mir and Breaz [38], and the inequalities of this result also provide an improvement of Theorem 1.7 due to Devi et al. [17] on dividing them by \(|\beta |\) and letting \(|\beta |\rightarrow \infty \). More precisely, we prove

### Theorem 2.15

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every complex number* *α* *and* *β* *with* \(|\alpha |<1\), \(|\beta |\geq 1\) *and for each* \(r>0\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.16

As mentioned earlier, for \(n\geq 2\), Theorem 2.15 amounts to the following remarkable result that improves Theorem 1.7 due to Devi et al. [17], the claim being followed from the same arguments as done in Remark 2.13.

### Corollary 2.17

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every complex number* *α* *with* \(|\alpha |<1\) *and for each* \(r>0\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.18

Further, letting \(r\rightarrow \infty \) on both sides of (45) and (46) of Corollary 2.17 and following the similar arguments of the results (30), (31), and (32) of Remark 2.6 to

and letting \(|\alpha |\rightarrow 1\), the following result follows that is an improvement of a result due to Aziz and Ahmad [3, Theorem 3].

### Corollary 2.19

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every complex number* *α* *and* *β* *with* \(|\alpha |<1\), \(|\beta |\geq 1\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.20

Taking \(\alpha =0\) and following the same arguments of Remark 2.18, Corollary 2.17 is aligned to the following which is a refinement of inequality (13) due to Govil [25].

### Corollary 2.21

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every complex number* *β* *with* \(|\beta |\geq 1\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.22

Similarly, as in Remark 2.18, first setting limit as \(r\rightarrow \infty \) on both sides of (43) and (44) of Theorem 2.17 and following the same arguments of the mentioned remark, we obtain

and

Letting \(|\alpha |\rightarrow 1\) on both sides of (49) and (50), we get a refinement of inequality (21) due to Mir and Breaz [38].

### Corollary 2.23

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every complex number* *β* *with* \(|\beta |\geq 1\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.24

Again, setting \(\alpha =0\) in inequalities (49) and (50) of Remark 2.22, we deduce a result that is an improvement of inequality (20) due to Mir and Breaz [38].

### Corollary 2.25

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* \(n\geq 2\) *having no zeros in* \(|z|< k\), \(k\leq 1\), *then for every complex number* *β* *with* \(|\beta |\geq 1\),

*and*

*provided* \(|p'(z)|\) *and* \(|q'(z)|\) *attain their maxima at the same point on the circle* \(|z|=1\).

### Remark 2.26

The study of special types of polynomials, particularly orthogonal polynomials and para-orthogonal polynomials on the unit circle, has received considerable interest from two disparate audiences, namely researchers in orthogonal polynomials and researchers in numerical linear algebra. On this subject, several recent papers (see, for instance, [9–12, 44], etc.) have been published which explore various properties of these polynomials, such as the interlacing of zeros using CMV (Cantero-Moral-Velázquez) matrices and necessary and sufficient conditions for monotonicity with a real parameter of the zeros of para-orthogonal polynomials. If one is interested in further studying the rate of change of such polynomials on the unit circle \(|z|=1\), then as some worthy consequences of our result (e.g., Theorem 2.1), we could have nice and remarkable additional information on the theory of orthogonal polynomials and para-orthogonal polynomials on the unit circle and is demonstrated as follows.

Here, we mainly follow the notation of [12]. Denote

Let \(C(a_{0}, a_{1},\ldots,a_{n-1}, b_{n})\) be the CMV matrix, where \(a_{j}\in \mathbf{D}\) and \(b_{n}\in \mathbf{S^{1}}\). The monic polynomial \(p_{n+1}\) defined by

is the para-orthogonal polynomial on the unit circle (POPUC) of degree \(n+1\) associated with the array \((a_{0}, a_{1},\ldots,a_{n-1}, b_{n})\). If \(b_{n} \in \mathbf{D}\), then \(p_{n+1}(z)\) is an orthogonal polynomial on the unit circle (OPUC) and all its zeros lie in **D** (see [12, Definition 1.1]).

Consider a CMV matrix \(C(a_{0},a_{1},b_{2})\) of order 3-by-3 defined by

where \(a_{0}\),\(a_{1}\), and \(b_{2}\) are in **D** and \(r_{i}=(1-|a_{i}|^{2})^{1/2}\) for \(i=0,1\).

Set \(\alpha _{11}=\bar{a_{0}}\), \(\alpha _{12}=r_{0}\bar{a_{1}}\), \(\alpha _{13}=r_{0}r_{1}\), \(\alpha _{21}= r_{0}\), \(\alpha _{22}=-a_{0}\bar{a_{1}}\), \(\alpha _{23}=-a_{0}r_{1}\), \(\alpha _{31}=0\), \(\alpha _{32}=\bar{b_{2}}r_{1}\) and \(\alpha _{33}=-\bar{b_{2}}a_{1}\).

Then, the characteristic polynomial

is an orthogonal polynomial on the unit circle of degree 3 with all its zeros in **D**.

Therefore, using the inequality (33) of Remark 2.6, we have

where \(\displaystyle{m_{1}=\min _{|z|=1}|p_{3}(z)|.}\)

Inequality (57) gives the lower bound estimates of \(\max _{|z|=1}|p_{3}'(z)|\) in terms of \(\max _{|z|=1}|p_{3}(z)|\) and \(m_{1}\).

Similarly, for an orthogonal polynomial of degree 2, we can easily discuss the bound of its derivative using the inequality (34) of the above-mentioned remark. Further, it is obvious that we can use the same inequalities for the bound of the derivative of para-orthogonal polynomials on the unit circle since all its zeros lie on the unit circle. Besides, we can also study the extremals properties of the polar derivative of the OPUC as well as that of POPUC using Corollary 2.10 of our result as well as their integral setting. Hence, these findings will pave a new direction for those working on orthogonal polynomials as well as para-orthogonal polynomials to extract interesting additional information concerning extremal properties.

## 3 Lemmas

In this section, we present some lemmas required to prove our results.

The first lemma is a special case of a result due to Govil and Rahman [26].

### Lemma 3.1

*If* \(p(z)\) *is a polynomial of degree* *n*, *then on* \(|z|=1\)

*where*

### Lemma 3.2

*If* \(p(z)\) *is a polynomial of degree* *n* *having no zeros in* \(|z|<1\), *then for every* \(R\geq 1\) *and every* \(r>0\),

*where*

Lemma 3.3 was proved by Boas and Rahman [7] for \(r\geq 1\) and by Rahman and Schmeisser [39] for \(0< r<1\). The next lemma is due to Frappier et al. [21].

### Lemma 3.3

*If* \(p(z)\) *is a polynomial of degree* *n*, *then for* \(R>1\),

*and*

### Lemma 3.4

*If* \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) *is a polynomial of degree* *n* *having all its zeros in* \(|z|\leq k\), \(k\leq 1\), *then for* \(|z|=1\),

## 4 Proof of the theorems

In this section, we demonstrate the proof of our main results.

### Proof of Theorem 2.1

Since \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) has all its zeros in \(|z|\leq k\), \(k\geq 1\), then for every real or complex number *α* with \(|\alpha |<1\), by Rouche’s theorem, the polynomial \( {R(z)=p(z)+\alpha m}\) has all its zeros in \(|z|\leq k\), \(k\geq 1\). Hence, the polynomial \(E(z)=R(kz)\) has all its zeros in \(|z|\leq 1\) and hence the polynomial \(F(z)=z^{n}\overline{E(\frac{1}{\overline{z}})}\) has all its zeros in \(|z|\geq 1\). If \(z_{\nu}\), \(\nu = 1, 2, 3,\ldots...,n\) are the zeros of \(F(z)\), then obviously \(|z_{\nu}|\geq 1\), \(1\leq \nu \leq n\) and

so that for points \(e^{i\theta}\), \(0\leq \theta < 2\pi \), for which \(F(e^{i\theta})\neq 0\), we have

which gives

for points \(e^{i\theta}\), \(0\leq \theta < 2\pi \), for which \(F\left (e^{i\theta}\right )\neq 0\).

Inequality (65) is equivalent to

for points \(e^{i\theta}\), \(0\leq \theta \leq 2\pi \), for which \(F\left (e^{i\theta}\right )\neq 0\). Inequality (66) trivially also holds for the points \(e^{i\theta}\), \(0\leq \theta <2\pi \), for which \(F\left (e^{i\theta}\right )=0\). Hence, it follows that for \(|z|=1\)

Since \(E(z)\) has all its zeros in \(|z|\leq 1\), by Gauss-Lucas theorem \(E'(z)\) has all its zeros in \(|z|\leq 1\), and hence the polynomial

has all its zeros in \(|z|\geq 1\).

From (68), it follows that the function

is analytic in \(|z|\leq 1\) with \(|W(z)|\leq 1\) for \(|z|\leq 1\) and \(W(0)=0\), hence the function \(1+W(z)\) is subordinate to the function \(1+z\) for \(|z|\leq 1\). Hence, by a well-known property of subordination [27], we have for every \(r>0\),

Now,

For \(|z|=1\), we have from (68)

For \(|z|=1\), relation (71) gives, on using equation (72),

Combining (70) and (73), we have for every \(r>0\)

Using inequality (59) of Lemma 3.2 to \(F(z)\), we get for each \(k\geq 1\) and every \(r>0\)

where

Since \(F(z)=z^{n}\overline{E\left (\dfrac{1}{\overline{z}}\right )}=z^{n} \overline{R\left (\dfrac{k}{\overline{z}}\right )}=z^{n}\bigg( \overline{p\bigg(\frac{k}{\overline{z}}\bigg)}+\overline{\alpha}m \bigg)\), we have for \(0\leq \theta < 2\pi \) and

From (74), (75) and (76), it follows that for every \(r>0\),

Applying Lemma 3.4 to \(E(z)\), we have for \(|z|=1\)

Now, using inequality (78), we have for \(|\frac{\beta}{k}|\geq 1\) and \(|z|=1\)

Replacing \(E(z)\) by \(R(kz)\) in (79), we get

which is equivalent to

so that we obtain

which is equivalent to

Applying inequality (80) to inequality (77)

Since \(D_{\beta}R(z)=D_{\beta}p(z)+\alpha mn\) is a polynomial of degree \(n-1\geq 1\), then using inequality (61) of Lemma 3.3, we have

which is equivalent to

which proves inequality (22).

The proof of inequality (23) follows on the same lines as that of inequality (26), but instead of using (61) of Lemma 3.3, we use inequality (62) of the same Lemma. □

### Proof of Theorem 2.15

Let \(p(z)=\displaystyle{\sum _{v=0}^{n}a_{v}z^{v}}\) be a polynomial of degree \(n\geq 2\) having no zero in \(|z|< k\), \(k\leq 1\).

Then, \(q(z)=z^{n}\overline{p\left (\frac{1}{\overline{z}}\right )}\) has all its zeros in \(|z|\leq 1/k \), \(1/k\geq 1\).

If

for every real or complex number *α* with \(|\alpha |< 1\), by Rouche’s theorem, the polynomial

where \(m=\min \limits _{|z|=k}|p(z)|\) has all its zeros in \(|z|\leq \frac{1}{k} \), \(\frac{1}{k}\geq 1\).

Applying Corollary 2.5 to the polynomial \(Q(z)\), we have for every \(r>0\)

and

which is equivalent to

and

By Lemma 3.1, we have for \(|z|=1\)

Since \(|p'(z)|\) and \(|q'(z)|\) attain their maxima at the same point on \(|z|=1\), let \(z_{0}\) on \(|z|=1\) be such that \(\displaystyle{\max \limits _{|z|=1}|q'(z)|=|q'(z_{0})|}\), then

Now, in particular, (86) gives

which implies

Applying (87) to (84) and (85), we have

and

From \(\displaystyle{q(z)=z^{n} \overline{p\left (\dfrac{1}{\overline{z}}\right )}}\), we have

On using equation (90) to (88) and (89), we have

and

Using Lemma 3.1, we have for \(|\beta |\geq 1\) and \(|z|=1\)

Using inequality (93) in inequalities (91) and (92), we have

and

which completes the proof of Theorem 2.15. □

### Proof of Theorem 2.11

The proof of this theorem follows on the same lines as that of Theorem 2.15 but instead of applying Corollary 2.5 to \(Q(z)\) given by (83), we simply apply the same lemma to \(q(z)=z^{n}\overline{p(\frac{1}{\overline{z}}),}\) and we omit it. □

## 5 Numerical examples and results

As an illustration of the obtained results, in this section, we consider the following examples and compare the values of the bounds obtained from our different results shown in the tables below.

### Example 5.1

Let \(p(z)=z^{3}-\frac {3}{2}z^{2}+\frac{1}{2}z\), with all zeros \(\{0.\frac{1}{2}, 1\}\) in \(|z|\leq \frac{3}{2}\) so that Corollary 2.10 holds for \(k\geq \frac{3}{2}\).

On the unit circle, we have

and its graphics for \(0\leq \theta <4\pi \) are presented in Fig. 1.

We first set \(\beta =3\in \mathbb{C}\) with \(|\beta |=3\), then

For \(\beta =2.5\in \mathbb{C}\) with \(|\beta |=2.5\), then

Again, \(\beta =2\in \mathbb{C}\) with \(|\beta |=2\), then

And for \(\beta =1.5\in \mathbb{C}\) with \(|\beta |=1.5\), then

Since

and for \(\beta =3,~2.5,~2\) and \(1.5\in \mathbb{C}\), we obtain

and

respectively.

We present the graphics for \(0\leq \theta <4\pi \) in Fig. 2 for \(D_{\beta}p(z)\) for some value of \(|\beta |\).

In Fig. 2, each of the graphs explains periodicity of 2*π* that can also be verified by substituting *θ* for \(2\pi -\theta \) in the value of \(|D_{\beta}p(e^{i\theta})|\) and also the interesting effects or variations of \(\displaystyle{\max _{0\leq \theta \leq 2\pi}|D_{\beta}p(e^{i\theta})|}\) occurring at \(\theta =\pi \) for \(|\beta |\) increasing that \(\displaystyle{\max _{0\leq \theta <2\pi}|D_{\beta}p(e^{i\theta})|}\) is stretching up most while for the \(\displaystyle{\min _{0\leq \theta \leq 2\pi}|D_{\beta}p(e^{i\theta})|}\) occurring, in particular, either at \(\theta =0\) or \(\theta =2\pi \), the stretching is least.

Now, we can consider the difference between the left-hand and right-hand sides in (37) of Corollary 2.10

For a fixed value of \(|\beta |=2\), graphic of the function \(k\mapsto \delta (k,\beta )\) for \(\frac{3}{2}\leq k\leq \frac{5}{2}\) is presented in Fig. 3. From the figure, it is evident that the least difference between the left-hand and right-hand sides of inequality (37) is obtained when the radius *k* of the closed disc \(|z|\leq k\) as zero region, is chosen a suitable value just exceeding the farthest zero of the polynomial from the origin.

For a fixed value of \(|\beta |=2\), in Fig. 4, we present graphics that compare the difference \(k\mapsto \phi (k)\) in inequality (37) of Corollary 2.10 and inequality (19) due to Aziz and Rather. In this graphic, we compare the difference \(k\mapsto \phi (k)\) between the left-hand and right-hand sides in the inequality (37) of Corollary 2.10 and (19) for Aziz and Rather for \(|\beta |=2\). From the graphics, it is clearly evident that our result (Corollary 2.10) yields a significantly better bound than that by Aziz and Rather mentioned above. Besides, very important information is clearly noticed from the graph that for obtaining the most accurate bound, the value of radius *k* of the disc \(|z|\leq k\) must be chosen as a suitable or convenient one as nearer as possible to the farthest zero of the polynomial from the origin.

In Table 1, we find the precentage in improvement of bound obtained from our results ( Corollary 2.10) compared to the bound from inequality (19) due Aziz and Rather for the polynomial \(p(z)=z^{3}-\frac {3}{2}z^{2}+\frac{1}{2}z\).

### Example 5.2

Let \(p(z)=(z+1)^{3}\), with its zeros \(\left \{-1,-1,-1\right \}\) in \(|z|\geq \frac{3}{4}\), so that Corollary 2.23 holds for \(k\leq \frac{3}{4}\).

On the unit circle, we have

and its graphic for \(0\leq \theta <4\pi \) is presented in Fig. 5.

We first take \(\beta =3.5\in \mathbb{C}\) with \(|\beta |=2\), then

Again for \(\beta =3\in \mathbb{C}\) with \(|\beta |=\), then

We set \(\beta =2\in \mathbb{C}\) with \(|\beta |=2\), then

Again, taking \(\beta =1.5\in \mathbb{C}\) with \(|\beta |=1.5\), then

And for \(\beta =1\in \mathbb{C}\) with \(|\beta |=1\), then

Since

and for \(\beta =3.5, 3, 2 , 1.5, 1\in \mathbb{C}\), we obtain

and

respectively.

Note that \(\displaystyle{m = \min _{|z|=k}|p(z)| = (1-k)^{3}}\) for \(k\leq \frac{3}{4}\).

We present the graphics for \(0\leq \theta <4\pi \) in Fig. 6 for \(D_{\beta}p(z)\) for some value of \(|\beta |=1, 1.5, 2, 2.5\), and 3.

In Fig. 6, each of the graphs explains the periodicity of 2*π*, which can also be verified by substituting *θ* for \(2\pi -\theta \) in the value of \(|D_{\beta}p(e^{i\theta})|\) and also the interesting effects or variations of \(\displaystyle{\max _{0\leq \theta \leq 2\pi}|D_{\beta}p(e^{i\theta})|}\) occurring either at \(\theta =0\) or \(\theta =2\pi \) for \(|\beta |\) increasing that \(\displaystyle{\max _{0\leq \theta \leq 2\pi}|D_{\beta}p(e^{i\theta})|}\) is stretching up most while for \(\displaystyle{\min _{0\leq \theta \leq 2\pi}|D_{\beta}p(e^{i\theta})|=0}\) occurring in particular at \(\theta =\pi \), there is no stretching.

Now, we can consider the difference between the left-hand and right-hand sides in (51) of Corollary 2.23

For a fixed value of \(|\beta |=2\), graphic of the function \(k\mapsto \gamma (k, \beta )\) for \(0\leq k\leq \frac{3}{4}\) is presented in Fig. 7.

From the graph, it is seen that if we set the value of *k* very near to the smallest modulus of the zeros, the bound better estimates \(\displaystyle{\max _{|z|=1}|D_{\beta}p(z)}|\).

In Fig. 8, we compare the difference \(k\mapsto \Delta (k)\) between the left-hand and right-hand sides in the inequalities (20) and (21) both due to Mir and Breaz [38] and inequality (51) of Corollary 2.23 for \(|\beta |=2\). From the graphics, it is obvious that our result (Corollary 2.23) yields a significantly better bound than that by Mir and Breaz mentioned above. Moreover, a very interesting point is clearly noticed from the graph that for obtaining the most accurate bound, the value of radius *k* of the zero free disc \(|z|< k\) must be chosen as a suitable or convenient one as nearer as possible to the shortest zero of the polynomial from the origin.

In Table 2, we present the precentage of improvement in the bound obtained from our results ( Corollary 2.23 and Corollary 2.25) compared to the bound from inequality (20) due to Mir Breaz [38] for the polynomial \(p(z)=(z+1)^{3}\).

## 6 Conclusion

This paper deals with the integral mean estimates of some well-known Turán-type as well as Erdös-Lax-type inequalities for the polar derivative of polynomials. The results obtained produce various inequalities that are sharper than the previous ones known in the very rich literature on this subject, where the authors have also verified the sharpness of their results by means of numerical examples and graphical illustrations.

## Data Availability

No datasets were generated or analysed during the current study.

## References

Arestov, V.V.: On inequalities for trigonometric polynomials and their derivative. Izv. Akad. Nauk SSSR, Ser. Mat.

**45**, 3–22 (1981)Aziz, A.: Integral mean estimates for polynomials with restricted zeros. J. Approx. Theory

**55**, 232–239 (1988)Aziz, A., Ahmad, N.: Inequalities for the derivative of a polynomial. Proc. Indian Acad. Sci. Math. Sci.

**107**(2), 189–196 (1997)Aziz, A., Dawood, Q.M.: Inequalities for a polynomial and its derivative. J. Approx. Theory

**54**, 306–313 (1988)Aziz, A., Rather, N.A.: A refinement of a theorem of Paul Turán concerning polynomial. Math. Inequal. Appl.

**1**, 231–238 (1998)Bernstein, S.: Lecons Sur Les Propriétés extrémales et la meilleure approximation des functions analytiques dúne fonctions reele Paris (1926)

Boas, R.P. Jr., Rahman, Q.I.: \(L^{p}\) inequalities for polynomials and entire functions. Arch. Ration. Mech. Anal.

**11**, 34–39 (1962)Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995)

Castillo, K.: On monotonicity of zeros of paraorthogonal polynomials on the unit circle. Linear Algebra Appl.

**580**, 475–490 (2019)Castillo, K.: Markov’s theorem for weight functions on the unit circle. Constr. Approx.

**55**, 605–627 (2022)Castillo, K., Cruz-Barroso, R., Perdomo-Pío, F.: On a spectral theorem in para-orthogonal theory. Pac. J. Math.

**208**, 71–91 (2016)Castillo, K., Petronilho, J.: Refined interlacing prpperties for zeros of paraorthogonal polynomials on the unit circle. Proc. Am. Math. Soc.

**146**, 3285–3294 (2018)Chan, T.N., Malik, M.A.: On Erdös-Lax theorem. Proc. Indian Acad. Sci. Math. Sci.

**92**(3), 191–193 (1983)Chanam, B., Dewan, K.K.: On integral mean estimates for polynomials. Int. J. Pure Appl. Math.

**41**, 797–805 (2007)Cheney, E.W.: Introduction to Approximation Theory. McGraw-Hill, New York (1966)

de-Bruijn, N.G.: Inequalities concerning polynomials in the complex domain. Nederl. Akad. Wetench. Proc. Ser. A

**50**, 1265–1272 (1947). Indag. Math. 9, 591–598 (1947)Devi, K.B., Krishnadas, K., Chanam, B.: \(L^{r}\) inequalities for the derivative of a polynomial. Note Mat.

**41**(2), 19–29 (2021)DeVore, R.A., Lorentz, G.G.: Constr. Approx. Springer, Berlin (1993)

Dewan, K.K., Bidkham, M.: Inequalities for a polynomial and its derivative. J. Math. Anal. Appl.

**166**, 319–324 (1992)Dewan, K.K., Mir, A.: On the maximum modulus of a polynomial and its derivatives. Int. J. Math. Math. Sci.

**16**, 2641–2645 (2005)Frappier, C., Rahman, Q.I., Rucheweyh, St.: New inequalities for polynomials. Trans. Am. Math. Soc.

**194**, 720–726 (1995)Gardner, R.B., Govil, N.K., Milovanović, G.V.: Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials. Mathematical Analysis and Its Applications. Elsevier, London (2022)

Gardner, R.B., Weems, A.: A Bernstein-type of \(L^{p}\) inequality for a certain class of polynomials. J. Math. Anal. Appl.

**219**, 472–478 (1998)Govil, N.K.: On the derivative of a polynomial. Proc. Am. Math. Soc.

**41**, 543–546 (1973)Govil, N.K.: On the theorem of S. Bernstein. Proc. Natl. Acad. Sci.

**50**, 50–52 (1980)Govil, N.K., Rahman, Q.I.: Functions of exponential type not vanishing in a half-plane and related polynomials. Trans. Am. Math. Soc.

**137**, 501–517 (1969)Hille, E.: Analytic Function Theory, Vol. II. Ginn. and Company, New York (1962)

Ivanov, V.I.: Some extremal properties of polynomials and inverse inequalities in approximatiom theory. Tr. Mat. Inst. Steklova

**145**, 79–110 (1979)Kumar, P.: On the inequalities concerning polynomials. Complex Anal. Oper. Theory

**14**(2020). Paper No. 65, 11 pp.Lax, P.D.: Proof of a conjecture of P. Erdös on the derivative of a polynomial. Bull. Am. Math. Soc.

**50**, 509–513 (1944)Lorentz, G.G.: Approximation of Functions. Holt, Rinehart and Winston, New York (1966)

Malik, M.A.: On the derivative of a polynomial. J. Lond. Math. Soc.

**1**, 57–60 (1969)Malik, M.A.: An integral mean estimates for polynomials. Proc. Am. Math. Soc.

**91**(2), 281–284 (1984)Marden, M.: Geometry of Polynomials. Mathematical Surveys, vol. 3. Am. Math. Soc., Providence (1966)

Markov, A.A.: On a problem of D.I. Mendeleev. Zap. Imp. Akad. Nauk, St. Petersburg

**62**, 1–24 (1889)Mendeleev, D.: Investigations of aqueous solutions based on specific gravity. St. Petersberg, 1887 (Russian)

Milovanović, G.V., Mitrinović, D.S., Rassias, Th.M.: Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

Mir, A., Breaz, D.: Bernstein and Turán-type inequalities for a polynomial with constraints on its zeros. RACSAM

**115**, 124 (2021)Rahman, Q.I., Schmeisser, G.: \(L^{p}\) inequalities for polynomials. J. Approx. Theory

**53**, 26–32 (1988)Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press, Oxford (2002)

Rather, N.A.: Extremal properties and location of the zeros of polynomials. Ph.D. Thesis, University of Kashmir (1998)

Rather, N.A., Bhat, F.A.: Inequalities for the polar derivative of a polynomial. Appl. Math. E-Notes

**17**, 231–241 (2017)Rudin, W.: Real and Complex Analysis. Mcgraw-Hill, New York (1977). (Reprinted in India)

Simon, B.: Orthogonal Polynomials on the Unit Circle, Part 1, Classical Theory, American Mathematical Society College Publications, vol. 54. Am. Math. Soc., Providence (2005)

Singha, N.K., Chanam, B.: On Turán-type integral mean estimate of a polynomial. Math. Found. Comput. (2023). https://doi.org/10.3934/mfc.2023048

Taylor, A.E.: Introduction to Functional Analysis. Wiley, New York (1958)

Telyakovski, S.A.: Research in the theory of approximation of functions at the mathematical institute of the academy of sciences. In: Proc. Steklov Inst. Math. Trudi Mat. Inst. Steklov, vol. 182, pp. 141–197 (1988). English trans, 1990

Tuŕan, P.: Über die anleitung von polymen. Compos. Math.

**7**, 89–95 (1939)Zygmund, A.: A remark on conjugate series. Proc. Lond. Math. Soc.

**34**, 392–400 (1932)

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Devi, K.B., Chanam, B. On Bernstein and Turán-type integral mean estimates for polar derivative of a polynomial.
*J Inequal Appl* **2024**, 107 (2024). https://doi.org/10.1186/s13660-024-03183-5

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DOI: https://doi.org/10.1186/s13660-024-03183-5