Generalizations of Zygmund-type integral inequalities for the polar derivative of a complex polynomial

Abstract

Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspired by the classical Bernstein-type inequalities that relate the uniform norms of polynomials and their derivatives on the unit circle, are investigated. The obtained results sharpen as well as generalize some already known $L^{\delta }$-estimates between polynomials and their polar derivatives.

1 Introduction and preliminaries

Let $P(z):=\sum_{\nu =0}^{n} a_{\nu }z^{\nu }$ be a complex polynomial of degree n and $P'(z)$ be its derivative. There is a well-known classical Bernstein result for two polynomials $P(z)$ and $Q(z)$ with degree of $P(z)$ not exceeding that of $Q(z)$ and $Q(z)\neq 0$ for $|z|>1$. Namely, the inequality $|P(z)|\leq |Q(z)|$ on the unit circle $|z|=1$ implies the inequality of their derivatives $|P^{\prime }(z)|\leq |Q^{\prime }(z)|$ on $|z|=1$. In particular, this result allows one to establish the famous Bernstein inequality [3] in the uniform norm on the unit circle, i.e., if $P(z)$ is a polynomial of degree n, it is true that

$$ \max_{ \vert z \vert =1} \bigl\vert {{P}^{\prime }}(z) \bigr\vert \le n \max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert . $$

(1.1)

This inequality (1.1) was proved by Bernstein in 1912, and is best possible with equality holding for monomials $P(z)= c z^{n}$, where c is an arbitrary complex number. If we restrict ourselves to the class of polynomials having no zeros in $|z|<1$, then inequality (1.1) can be sharpened. In fact, Erdős conjectured and later Lax [8] proved that, if $P(z)\neq 0$ in $|z|<1$, then

$$ \max_{ \vert z \vert =1} \bigl\vert {{P}^{\prime }}(z) \bigr\vert \le \frac{n}{2} \max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert . $$

(1.2)

In 1969 Malik [9] proved an extension of (1.2) under the condition that $P(z)\neq 0$ in $|z|< k$, where $k\geq 1$, i.e.,

$$ \max_{ \vert z \vert =1} \bigl\vert {{P}^{\prime }}(z) \bigr\vert \le \frac{n}{1+k}\max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert . $$

(1.3)

This inequality was generalized in different directions. Chan and Malik [4] proved that, if $P(z)=a_{0} +\sum_{j=\mu }^{n}a_{j} z^{j}$, $\mu \geq 1$, is a lacunary polynomial of degree n and $P(z)\neq 0$ in $|z|< k$, $k\geq 1$, then

$$ \max_{ \vert z \vert =1} \bigl\vert {{P}^{\prime }}(z) \bigr\vert \le \frac{n}{1+k^{\mu }}\max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert . $$

(1.4)

Inequality (1.4) was independently proved by Qazi (see [17], Lemma 1), who also under the same conditions proved that

$$ \max_{ \vert z \vert =1} \bigl\vert {{P}^{\prime }}(z) \bigr\vert \le \frac{n}{1+S_{0}(\mu )}\max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert , $$

(1.5)

where

$$ S_{0}(\mu )=k^{\mu +1} \biggl( \frac{ (\frac{\mu }{n} ) \vert \frac{a_{\mu }}{a_{0}} \vert k^{\mu -1}+1}{ (\frac{\mu }{n} ) \vert \frac{a_{\mu }}{a_{0}} \vert k^{\mu +1}+1} \biggr) \ge k^{\mu }. $$

(1.6)

For a polynomial $P(z)$ of degree n, now we define the so-called the polar derivative of $P(z)$ with respect to the point α as

$$ D_{\alpha }P(z):=nP(z)+(\alpha -z)P^{\prime }(z). $$

This polynomial $D_{\alpha }P(z)$ is of degree at most $n-1$ and it generalizes the ordinary derivative $P'(z)$ in the following sense:

$$ \lim_{\alpha \to \infty } \biggl\{ \frac{D_{\alpha }P(z)}{\alpha } \biggr\} =P'(z), $$

uniformly with respect to z for $|z|\leq R$ and $R>0$.

Aziz [1] was among the first who expanded some of the above inequalities by replacing the ordinary derivative with polar polynomial derivatives. In fact, In 1988, Aziz proved that, if $P(z)$ is a polynomial of degree n and $P(z)\neq 0$ in $|z|< k$, with $k\geq 1$, then for each complex number α such that $|\alpha |\geq 1$,

$$ \max_{ \vert z \vert =1} \bigl\vert D_{\alpha }P(z) \bigr\vert \le n \biggl(\frac{ \vert \alpha \vert +k}{1+k} \biggr)\max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert . $$

(1.7)

Dividing both sides of inequality (1.7) by $|\alpha |$ and letting $|\alpha |\rightarrow \infty$, we obtain (1.3).

Over the last four decades many different authors produced a large number of results pertaining to the polar derivative of polynomials. More information on this topic can be found in the books of Milovanović et al. [11], Rahman and Schmeisser [18], and Marden [10]. One can also see in the literature (for example, refer [6, 7, 12, 13, 15, 16, 19]) the latest research and development in this direction. Recently, Mir [12] extended (1.7) to the class of lacunary type polynomials $P(z)=a_{0}+\sum_{\nu =\mu }^{n}a_{\nu }z^{\nu }$, $\mu \ge 1$, not vanishing in $|z|< k$, $k\ge 1$, by first proving an $L^{\delta }$-norm estimate of the polar derivative $D_{\alpha }P(z)$ with $|\alpha |\ge 1$ and then from the result so obtained produced the desired generalization of (1.7). More precisely, Mir proved the following result.

Theorem 1.1

(cf. [12])

If$P(z)=a_{0}+\sum_{\nu =\mu }^{n}a_{\nu }z^{\nu }$, $1\le \mu \le n$, is a lacunary polynomial of degreenhaving no zero in$|z|< k$, where$k\ge 1$, then for every$\delta >0$and for every complex numberαsuch that$|\alpha |\ge 1$,

$$ \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha }P \bigl({\mathrm {e}}^{{\mathrm {i}} \theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}} \theta \biggr\} ^{\frac{1}{\delta }} \le n \bigl( \vert \alpha \vert +k^{\mu } \bigr)C_{\delta }\bigl(k^{\mu }\bigr) \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}} \theta \biggr\} ^{\frac{1}{\delta }}, $$

(1.8)

where

$$ C_{\delta }\bigl(k^{\mu }\bigr)= \biggl\{ \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert k^{\mu }+{ \mathrm {e}}^{\mathrm {i}\beta } \bigr\vert ^{\delta }{\,\mathrm {d}}\beta \biggr\} ^{- \frac{1}{\delta }}. $$

(1.9)

If we let $\delta \rightarrow \infty$ in (1.8), noting that $C_{\delta }(k^{\mu })\rightarrow {1}/({1+k^{\mu }})$, we get

$$ \max_{ \vert z \vert =1} \bigl\vert D_{\alpha }P(z) \bigr\vert \le n \biggl( \frac{ \vert \alpha \vert +k^{\mu }}{1+k^{\mu }} \biggr) \max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert , $$

(1.10)

which clearly represents a generalization of (1.7). In the same paper, Mir [12] further generalized inequalities (1.8) and (1.10) by proving the following result.

Theorem 1.2

(cf. [12])

If$P(z)= z^{s} (a_{0}+\sum_{j=\mu }^{n-s}a_{j}{z}^{j} )$, $1\le \mu \le n-s$, $0\le s \le n-1$, is a polynomial of degreenhavings-fold zeros at the origin and the remaining$n-s$zeros in$|z|\ge k$, $k\geq 1$, then for every complex numberαsuch that$|\alpha |\geq 1$,

$$ \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha }P \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}\le \bigl\{ (n-s) \bigl( \vert \alpha \vert +k^{\mu }\bigr)C_{\delta }\bigl(k^{\mu }\bigr)+s \vert \alpha \vert \bigr\} \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}, $$

(1.11)

where$C_{\delta }(k^{\mu })$is as defined in (1.9).

Making $\delta \rightarrow \infty$ in (1.11), we get

$$ \max_{ \vert z \vert =1} \bigl\vert D_{\alpha }P(z) \bigr\vert \le \biggl( \frac{n( \vert \alpha \vert +k^{\mu })+s( \vert \alpha \vert -1)k^{\mu }}{1+k^{\mu }} \biggr) \max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert , $$

(1.12)

which clearly is a generalization of (1.10).

The main aim of this paper is to further generalize and strengthen all the previous mentioned inequalities by establishing some general Zygmund-type integral inequalities for polynomials involving the polar derivative. The obtained results represent polar derivative analogues of some classical Bernstein-type inequalities in the uniform norm on the unit circle and include also several interesting generalizations and refinements of some $L^{\delta }$-norm inequalities for polynomials.

2 Main results

Here, we first prove the following more general inequality which includes not only (1.8) as a special case, but also derived polar derivative analogues of some classical Bernstein-type inequalities that relate the uniform norm of a polynomial to that of its derivative on the unit disk.

Theorem 2.1

If$P(z)= a_{0} +\sum_{j=\mu }^{n}a_{j}z^{j}$, $1\leq \mu \leq n$, is a lacunary polynomial of degreensuch that$P(z)\neq 0$in$|z|< k$, where$k\geq 1$, then for all complex numbersαandβ, with$|\alpha |\geq 1$and$\delta \geq 1$,

$$\begin{aligned} & \biggl\{ \int _{0}^{2\pi } \biggl\vert {\mathrm {e}}^{\mathrm {i}\theta }D_{ \alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+n\beta \biggl( \frac{ \vert \alpha \vert -1}{1+S_{0}(\mu )} \biggr)P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \biggr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} \\ &\quad \leq n \bigl[\bigl( \vert \alpha \vert +S_{0}(\mu )\bigr)+ \vert \beta \vert \bigl( \vert \alpha \vert -1\bigr) \bigr] C_{\delta }\bigl(S_{0}(\mu )\bigr) \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({ \mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta } {\,\mathrm {d}}\theta \biggr\} ^{ \frac{1}{\delta }}, \end{aligned}$$

(2.1)

where

$$ C_{\delta }\bigl(S_{0}(\mu )\bigr)= \biggl\{ \frac{1}{2\pi } \int _{0}^{2 \pi } \bigl\vert S_{0}( \mu )+{\mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta }{\,\mathrm {d}}t \biggr\} ^{-\frac{1}{\delta }} $$

(2.2)

and$S_{0}(\mu )$is defined by formula (1.6).

In the limiting case, when $\delta \rightarrow \infty$, the result is best possible for $\beta =0$ and equality in (2.1) holds for $P(z)=(z^{\mu }+k^{\mu })^{n/\mu }$, where n is a multiple of μ and $\alpha \ge 1$. Setting $\mu =1$ in Theorem 2.1, we obtain the following result.

Corollary 2.2

If$P(z)=\sum_{j=0}^{n}a_{j}z^{j}$is a lacunary polynomial of degreensuch that$P(z)\neq 0$in$|z|< k$, where$k\geq 1$, then for all complex numbersαandβ, with$|\alpha |\geq 1$and$\delta \geq 1$,

$$\begin{aligned} & \biggl\{ \int _{0}^{2\pi } \biggl\vert {\mathrm {e}}^{\mathrm {i}\theta }D_{ \alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+n\beta \biggl( \frac{ \vert \alpha \vert -1}{1+S_{1}} \biggr)P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \biggr\vert ^{\delta }{ \,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} \\ &\quad \leq n \bigl[\bigl( \vert \alpha \vert +S_{1}\bigr)+ \vert \beta \vert \bigl( \vert \alpha \vert -1\bigr) \bigr] B_{\delta } \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr) \bigr\vert ^{\delta } {\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}, \end{aligned}$$

(2.3)

where

$$ S_{1}=k^{2} \biggl( \frac{ \vert \frac{a_{1}}{na_{0}} \vert +1}{ \vert \frac{a_{1}}{na_{0}} \vert k^{2}+1} \biggr) $$

and

$$ B_{\delta }= \biggl\{ \frac{1}{2\pi } \int _{0}^{2\pi } \bigl\vert S_{1}+{ \mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta }{\,\mathrm {d}}t \biggr\} ^{-\frac{1}{\delta }}. $$

Remark 2.3

For $k=1$, we have $S_{1}=1$ and Corollary 2.2 reduces to a result of Mir and Wani [16] for $\delta \geq 1$. Dividing the two sides of (2.3) by $|\alpha |$ and letting $|\alpha |\to \infty$, we get the following result.

Corollary 2.4

If$P(z)=\sum_{j=0}^{n}a_{j}z^{j}$is a lacunary polynomial of degreensuch that$P(z)\neq 0$in$|z|< k$, where$k\geq 1$, then for each complex numberβand$\delta \geq 1$,

$$ \biggl\{ \int _{0}^{2\pi } \biggl\vert {\mathrm {e}}^{\mathrm {i}\theta }P'\bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr)+ \frac{n\beta }{1+S_{1}}P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \biggr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}\leq n \bigl( \vert \beta \vert +1\bigr) B_{ \delta } \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}, $$

(2.4)

where$S_{1}$and$B_{\delta }$are as defined in Corollary 2.2.

Taking $\beta =0$ in Theorem 2.1, we obtain the following result.

Corollary 2.5

If$P(z)= a_{0} +\sum_{j=\mu }^{n}a_{j}{z}^{j}$, $1\leq \mu \leq n$, is a lacunary polynomial of degreenhaving no zeros in$|z|< k$, $k\geq 1$, then for each complex numberαsuch that$|\alpha |\geq 1$and$\delta \geq 1$, we have

$$ \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha }P \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}\leq n\bigl( \vert \alpha \vert +S_{0}( \mu ) \bigr) C_{\delta }\bigl(S_{0}(\mu )\bigr) \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}, $$

(2.5)

where$S_{0}(\mu )$is defined by formula (1.6) and$C_{\delta }(S_{0}(\mu ))$is as defined by (2.2).

In the limiting case, when $\delta \rightarrow \infty$, the result is best possible and equality in (2.5) holds for $P(z)=(z^{\mu }+k^{\mu })^{n/\mu }$, where n is a multiple of μ and $\alpha \ge 1$.

Remark 2.6

If $P(z)= a_{0} +\sum_{j=\mu }^{n}a_{j}{z}^{j}\neq 0$ in $|z|< k$, $k\geq 1$, then by Lemma 3.5 for $s=0$, we have $\psi _{0}(\mu )=S_{0}(\mu )\geq k^{\mu }$. Taking $a=|\alpha |\geq 1$, $b=S_{0}(\mu )$, $c=k^{\mu }$ in Lemma 3.3, we obtain the following inequality:

$$ \frac{ \vert \alpha \vert +S_{0}(\mu )}{ \{\int _{0}^{2\pi } \vert S_{0}(\mu )+{\mathrm {e}}^{{\mathrm {i}}\beta } \vert ^{\delta }{\,\mathrm {d}} \beta \}^{1/\delta }} \leq \frac{ \vert \alpha \vert +k^{\mu }}{ \{\int _{0}^{2\pi } \vert {\mathrm {e}}^{{\mathrm {i}}\beta }+k^{\mu } \vert ^{\delta }{\,\mathrm {d}}\beta \}^{1/\delta }}, $$

which holds for each $\delta \geq 1$. This shows that Corollary 2.5 sharpens the bound in (1.8).

Remark 2.7

Inequality (2.5) has also been recently established by Kumar [7].

Dividing both the sides of (2.5) by $|\alpha |$ and letting $|\alpha |\rightarrow \infty$, we get Theorem 2.1 of Gardner and Weems [5] (see also Mir, Dewan, and Singh [14]).

It is important to mention that inequalities involving polynomials in the uniform norm on the unit circle in the complex plane are special cases of the polynomial inequalities involving the integral norm. For example, if we let $\delta \rightarrow \infty$ in (2.5), noting that $C_{\delta }(S_{0}(\mu ))\rightarrow {1}/(1+S_{0}(\mu ))$, we get the following refinement of (1.10).

Corollary 2.8

If$P(z)= a_{0} +\sum_{j=\mu }^{n}a_{j}{z}^{j}$, $1\leq \mu \leq n$, is a lacunary polynomial of degreenhaving no zeros in$|z|< k$, $k\geq 1$, then for each complex numberαsuch that$|\alpha |\geq 1$, we have

$$ \max_{ \vert z \vert =1} \bigl\vert D_{\alpha }P(z) \bigr\vert \leq n \biggl( \frac{ \vert \alpha \vert +S_{0}(\mu )}{1+S_{0}(\mu )} \biggr) \max_{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert , $$

(2.6)

where$S_{0}(\mu )$is as defined in formula (1.6).

As an application of Corollary 2.5, we shall also prove the following result which provides a generalization of (1.11) and many other related results.

Theorem 2.9

If$P(z)= z^{s} (a_{0}+\sum_{j=\mu }^{n-s}a_{j}{z}^{j} )$, with$0\le s \le n-1$and$1\le \mu \le n-s$, is a lacunary polynomial of degreenhavings-fold zeros at the origin and the remaining$n-s$zeros in$|z|\ge k$, where$k\geq 1$, then for every complex numberαsuch that$|\alpha |\geq 1$and$\delta \geq 1$, the following inequality

$$\begin{aligned} & \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha }P \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} \\ &\quad \leq \bigl[(n-s) \bigl( \vert \alpha \vert +\psi _{s}(\mu ) \bigr)C_{\delta }\bigl( \psi _{s}(\mu )\bigr)+s \vert \alpha \vert \bigr] \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({ \mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta } {\,\mathrm {d}}\theta \biggr\} ^{ \frac{1}{\delta }} \end{aligned}$$

(2.7)

holds, where

$$ \psi _{s}(\mu )=k^{\mu +1} \biggl\{ \frac{ (\frac{\mu }{n-s} ) \vert \frac{a_{\mu }}{a_{0}} \vert k^{\mu -1}+1}{ (\frac{\mu }{n-s} ) \vert \frac{a_{\mu }}{a_{0}} \vert k^{\mu +1}+1} \biggr\} $$

and

$$ C_{\delta }\bigl(\psi _{s}(\mu )\bigr)= \biggl\{ \frac{1}{2\pi } \int _{0}^{2 \pi } \bigl\vert \psi _{s}(\mu )+{\mathrm {e}}^{{\mathrm {i}}\beta } \bigr\vert ^{\delta }{\,\mathrm {d}} \beta \biggr\} ^{-\frac{1}{\delta }}. $$

In the limiting case, when$\delta \rightarrow \infty$, the result is best possible for$s=0$and equality in (2.7) holds for$P(z)=(z^{\mu }+k^{\mu })^{n/\mu }$, wherenis a multiple ofμand$\alpha \ge 1$.

Remark 2.10

For $s=0$, Theorem 2.9 reduces to Corollary 2.5.

Remark 2.11

By Lemma 3.5, we have $\psi _{s}(\mu )\geq k^{\mu }$. Taking $a=|\alpha |\geq 1$, $b=\psi _{s}(\mu )$, $c=k^{\mu }$ in Lemma 3.3, we get, for each $\delta \geq 1$,

$$ \frac{ \vert \alpha \vert +\psi _{s}(\mu )}{ \{ \int _{0}^{2\pi } \vert \psi _{s}(\mu )+{\mathrm {e}}^{{\mathrm {i}}\beta } \vert ^{\delta }{\,\mathrm {d}}\beta \} ^{1/\delta }} \leq \frac{ \vert \alpha \vert +k^{\mu }}{ \{ \int _{0}^{2\pi } \vert k^{\mu }+{\mathrm {e}}^{{\mathrm {i}}\beta } \vert ^{\delta }{\,\mathrm {d}}\beta \} ^{1/\delta }}. $$

This shows that Theorem 2.9 sharpens the bound in (1.11).

Remark 2.12

Making $\delta \rightarrow \infty$ in (2.7), we get

$$ \max_{ \vert z \vert =1} \bigl\vert D_{\alpha }P(z) \bigr\vert \le \biggl( \frac{n( \vert \alpha \vert +\psi _{s}(\mu ))+s( \vert \alpha \vert -1)\psi _{s}(\mu )}{1+\psi _{s}(\mu )} \biggr) \max _{ \vert z \vert =1} \bigl\vert P(z) \bigr\vert , $$

(2.8)

where $\psi _{s}(\mu )$ is as defined in Theorem 2.9. It is easy to verify that, for every complex number α with $|\alpha |\ge 1$ and $k\ge 1$, the function

$$ x\mapsto \frac{n( \vert \alpha \vert +x)+s( \vert \alpha \vert -1)x}{1+x},\quad x\ge 0, $$

is a nonincreasing function in x. If we combine this fact with Lemma 3.5, according to which $\psi _{s}(\mu )\ge k^{\mu }$ for $\mu \ge 1$, we find that the right-hand side of (2.8) does not exceed the right-hand side of (1.12). Thus (2.8) represents a refinement of (1.12).

3 Auxiliary results

In order to prove our main results (Theorems 2.1 and 2.9), we need the following lemmas.

Lemma 3.1

Let$P(z)= a_{0} +\sum_{\nu =\mu }^{n}a_{\nu }{z}^{\nu }$, $\mu \geq 1$, be a lacunary polynomial of degreenwithout zeros in the disc$|z|< k$, where$k\geq 1$. Then on$|z|=1$,

$$ S_{0}(\mu ) \bigl\vert P^{\prime }(z) \bigr\vert \leq \bigl\vert Q^{\prime }(z) \bigr\vert $$

(3.1)

and

$$ \frac{\mu }{n} \biggl\vert \frac{a_{\mu }}{a_{0}} \biggr\vert k^{\mu }\le 1, $$

(3.2)

where$Q(z)=z^{n}\overline{P({1}/{\bar{z}})}$and$S_{0}(\mu )$is as defined by (1.6).

The above lemma is implicit in Qazi [17].

The next two lemmas have been recently proved by Govil and Kumar [6].

Lemma 3.2

Ifpandqare arbitrary positive real numbers such that$q\ge px$, where$x\geq 1$, and ifβis any real number such that$0\leq \beta < 2\pi$, then

$$ \frac{q+py}{x+y}\leq \biggl\vert \frac{q+p{\mathrm {e}}^{{\mathrm {i}}\beta }}{x+{\mathrm {e}}^{{\mathrm {i}}\beta }} \biggr\vert $$

for each$y\geq 1$.

Lemma 3.3

For real numbers$a\geq 1$, $b\geq c\geq 1$, and$\delta > 0$, we have the following inequality:

$$ \frac{a+b}{ \{ \int _{0}^{2\pi } \vert {\mathrm {e}}^{\mathrm {i}\theta }+b \vert ^{\delta }{\,\mathrm {d}}\theta \} ^{1/\delta }} \leq \frac{a+c}{ \{ \int _{0}^{2\pi } \vert {\mathrm {e}}^{\mathrm {i}\theta }+c \vert ^{\delta }{\,\mathrm {d}}\theta \} ^{1/\delta }}. $$

We need also the following lemma proved by Aziz and Shah [2].

Lemma 3.4

Let$P(z)$be a polynomial of degreenand$Q(z)$be defined by$Q(z)=z^{n}\overline{P({1}/{\bar{z}})}$. Then, for every$\delta >0$, the following inequality

$$ \int _{0}^{2\pi } \int _{0}^{2\pi }\bigl|Q^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+{ \mathrm {e}}^{{\mathrm {i}}\varphi }P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|^{\delta }{\,\mathrm {d}} \varphi {\,\mathrm {d}}\theta \leq 2\pi n^{\delta } \int _{0}^{2\pi }\bigl|P\bigl({ \mathrm {e}}^{\mathrm {i}\theta } \bigr)\bigr|^{\delta }{\,\mathrm {d}}\theta $$

holds.

Lemma 3.5

Let$P(z)$be a lacunary polynomial of degreenhavings-fold zeros at the origin and the remaining$n-s$zeros in$|z|\ge k$, where$k\geq 1$, i.e.,

$$ P(z)= z^{s} \Biggl(a_{0}+\sum _{j=\mu }^{n-s}a_{j}{z}^{j} \Biggr),\quad 0\le s \le n-1, 1\le \mu \le n-s. $$

Then

$$ \psi _{s}(\mu )=k^{\mu +1} \biggl\{ \frac{ (\frac{\mu }{n-s} ) \vert \frac{a_{\mu }}{a_{0}} \vert k^{\mu -1}+1}{ (\frac{\mu }{n-s} ) \vert \frac{a_{\mu }}{a_{0}} \vert k^{\mu +1}+1} \biggr\} \ge k^{\mu }. $$

Proof

We put $P(z)= z^{s}\phi (z)$, with

$$ \phi (z)= a_{0}+\sum_{\nu =\mu }^{n-s}a_{\nu }z^{\nu }, $$

and suppose that $\phi (z) \neq 0$ in the disc $|z|< k$, where $k\ge 1$.

Then on applying inequality (3.2) of Lemma 3.1 to the polynomial $\phi (z)$ of degree $n-s$, we get

$$ \biggl(\frac{\mu }{n-s} \biggr) \biggl\vert \frac{a_{\mu }}{a_{0}} \biggr\vert k^{\mu }\le 1, $$

which is equivalent to

$$ \biggl(\frac{\mu }{n-s} \biggr) \biggl\vert \frac{a_{\mu }}{a_{0}} \biggr\vert k^{\mu }(k-1)\le k-1, $$

which implies

$$ \biggl(\frac{\mu }{n-s} \biggr) \biggl\vert \frac{a_{\mu }}{a_{0}} \biggr\vert k^{ \mu +1}+1\le \biggl(\frac{\mu }{n-s} \biggr) \biggl\vert \frac{a_{\mu }}{a_{0}} \biggr\vert k^{\mu }+k, $$

from which Lemma 3.5 follows. □

4 Proofs of main results

Proof of Theorem 2.1

By hypothesis we have $P(z)=a_{0}+\sum_{j=\mu }^{n}a_{j}z^{j}\ne 0$ in $|z|< k$, where $k\geq 1$. Supposing $Q(z)=z^{n}\overline{P({1}/{\bar{z}})}$, we conclude that $P(z)=z^{n}\overline{Q({1}/{\bar{z}})}$ and it can be easily checked that, for each θ ($0\leq \theta < 2\pi$),

$$ nP\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)-{\mathrm {e}}^{\mathrm {i}\theta }P^{\prime } \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)={ \mathrm {e}}^{{\mathrm {i}}(n-1)\theta } \overline{Q^{\prime } \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)}, $$

as well as for polar derivative

$$ D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)=nP\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+\bigl(\alpha -{\mathrm {e}}^{ \mathrm {i}\theta } \bigr)P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr), $$

where $\alpha \in \mathbb{C}$. Using these equalities, we obtain

$$\begin{aligned} \bigl|D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|&\leq \bigl|nP\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)-{\mathrm {e}}^{ \mathrm {i}\theta }P^{\prime } \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert + \vert \alpha \vert \bigl\vert P^{\prime }\bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr)\bigr| \\ &=\bigl|Q^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert + \vert \alpha \vert \bigl\vert P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|. \end{aligned}$$

(4.1)

Now, for $\delta \geq 1$, using Minkowski’s inequality, we get

$$\begin{aligned} &\biggl\{ \int _{0}^{2\pi } \bigl\vert S_{0}( \mu ) +{\mathrm {e}}^{{ \mathrm {i}}t} \bigr\vert ^{\delta }{\,\mathrm {d}}t \int _{0}^{2\pi } \biggl\vert {\mathrm {e}}^{\mathrm {i}\theta }D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+n\beta \biggl( \frac{ \vert \alpha \vert -1}{1+S_{0}(\mu )} \biggr)P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \biggr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} \\ &\quad = \biggl\{ \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert S_{0}( \mu ) +{ \mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta } \\ &\qquad {} \times \biggl\vert {\mathrm {e}}^{\mathrm {i}\theta }D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+n\beta \biggl(\frac{ \vert \alpha \vert -1}{1+S_{0}(\mu )} \biggr)P\bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr) \biggr\vert ^{\delta }{\,\mathrm {d}}\theta {\,\mathrm {d}}t \biggr\} ^{ \frac{1}{\delta }} \\ &\quad \leq \biggl\{ \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert S_{0}( \mu ) +{ \mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta } \bigl\vert D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta {\,\mathrm {d}}t \biggr\} ^{\frac{1}{\delta }} \\ &\qquad {} +n \vert \beta \vert \bigl( \vert \alpha \vert -1\bigr) \biggl\{ \int _{0}^{2\pi } \int _{0}^{2\pi } \biggl\vert \frac{S_{0}(\mu )+{\mathrm {e}}^{{\mathrm {i}}t}}{1+S_{0}(\mu )} \biggr\vert ^{\delta } \bigl\vert P\bigl({ \mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta {\,\mathrm {d}}t \biggr\} ^{\frac{1}{\delta }} \\ &\quad \leq \biggl\{ \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert S_{0}( \mu ) +{ \mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta } \bigl\vert D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta {\,\mathrm {d}}t \biggr\} ^{\frac{1}{\delta }} \\ &\qquad {} +n \vert \beta \vert \bigl( \vert \alpha \vert -1\bigr) (2\pi )^{1/\delta } \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}} \theta \biggr\} ^{\frac{1}{\delta }}. \end{aligned}$$

(4.2)

Recall that $P(z)\neq 0$ in $|z|< k$, where $k\geq 1$, we have by inequality (3.1) of Lemma 3.1 for $0\le \theta <2\pi$,

$$ S_{0}(\mu ) \bigl|P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr| \leq \bigl|Q^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|. $$

This gives, by taking $p=|P'({\mathrm {e}}^{\mathrm {i}\theta })|$, $q=|Q'({\mathrm {e}}^{\mathrm {i}\theta })|$, and $x=S_{0}(\mu )$ in Lemma 3.2, for each complex number α with $y=|\alpha |\ge 1$,

$$\begin{aligned} &\bigl( \bigl\vert Q^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert + \vert \alpha \vert \bigl\vert P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert \bigr) \bigl\vert S_{0}(\mu ) +{\mathrm {e}}^{{\mathrm {i}}t} \bigr\vert \\ &\quad \leq \bigl(S_{0}(\mu ) + \vert \alpha \vert \bigr) \bigl\vert \bigl\vert Q^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert +{ \mathrm {e}}^{{\mathrm {i}}t} \bigl\vert P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta } \bigr) \bigr\vert \bigr\vert . \end{aligned}$$

(4.3)

On applying (4.1) and (4.3), we get, for each $\delta \geq 1$,

$$\begin{aligned} &\int _{0}^{2\pi } \bigl\vert S_{0}( \mu ) +{\mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta }{ \,\mathrm {d}}t \int _{0}^{2\pi }\bigl|D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|^{\delta }{\,\mathrm {d}}\theta \\ &\quad = \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert S_{0}( \mu )+{\mathrm {e}}^{{ \mathrm {i}}t} \bigr\vert ^{\delta }\bigl|D_{\alpha }P \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|^{\delta }{\,\mathrm {d}}t {\,\mathrm {d}}\theta \\ &\quad \leq \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert S_{0}( \mu )+{\mathrm {e}}^{{ \mathrm {i}}t} \bigr\vert ^{\delta } \bigl[ \bigl\vert Q'\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert + \vert \alpha \vert \bigl\vert P'\bigl({ \mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert \bigr]^{\delta }{\,\mathrm {d}}t {\,\mathrm {d}}\theta \\ &\quad \leq \bigl(S_{0}(\mu )+ \vert \alpha \vert \bigr)^{\delta } \int _{0}^{2\pi } \int _{0}^{2 \pi } \bigl\vert \bigl\vert Q' \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert +{\mathrm {e}}^{{\mathrm {i}}t} \bigl\vert P'\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert \bigr\vert ^{\delta }{\,\mathrm {d}}t {\,\mathrm {d}}\theta . \end{aligned}$$

(4.4)

Since for each $\delta \geq 1$ and arbitrary $a,b\in \mathbb{C}$ the equality

$$ \int _{0}^{2\pi } \bigl\vert a+{\mathrm {e}}^{{\mathrm {i}}t}b \bigr\vert ^{\delta }{\,\mathrm {d}}t = \int _{0}^{2 \pi }\bigl| \vert a \vert +{\mathrm {e}}^{{\mathrm {i}}t}|b \vert \bigr\vert ^{\delta }{\,\mathrm {d}}t $$

holds, using Lemma 3.4 for each $\delta \geq 1$ and $|\alpha |\geq 1$, we obtain

$$\begin{aligned} &\int _{0}^{2\pi } \bigl\vert S_{0}( \mu ) +{\mathrm {e}}^{{\mathrm {i}}t} \bigr\vert ^{\delta }{ \,\mathrm {d}}t \int _{0}^{2\pi }\bigl|D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|^{\delta }{\,\mathrm {d}}\theta \\ &\quad \leq \bigl(S_{0}(\mu ) + \vert \alpha \vert \bigr)^{\delta } \int _{0}^{2\pi } \int _{0}^{2\pi }\bigl|Q^{\prime }\bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr)+{\mathrm {e}}^{{\mathrm {i}}t}P^{\prime }\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|^{\delta }{\,\mathrm {d}}t {\,\mathrm {d}}\theta \\ &\quad \leq \bigl(S_{0}(\mu ) + \vert \alpha \vert \bigr)^{\delta }2\pi n^{\delta } \int _{0}^{2\pi }\bigl|P\bigl({\mathrm {e}}^{\mathrm {i}\theta } \bigr)\bigr|^{\delta }{\,\mathrm {d}}\theta , \end{aligned}$$

which on raising the power ${1}/{\delta }$ on both sides and then using in (4.2) gives (2.1), and this completes the proof of Theorem 2.1. □

Proof of Theorem 2.9

As in Lemma 3.5, we put $P(z)= z^{s} \phi (z)$, where

$$ \phi (z)={{a}_{0}}+\sum_{j=\mu }^{n-s}{{{a}_{j}} {{z}^{j}}},\quad 1 \leq \mu \leq n-s, $$

is a polynomial of degree $n-s$ having no zeros in $|z|< k$, where $k\geq 1$. Applying Corollary 2.5 to the polynomial $\phi (z)$ of degree $n-s$, we obtain, for $|\alpha |\geq 1$ and $\delta \geq 1$,

$$ \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha } \phi \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} \leq \frac{(n-s)( \vert \alpha \vert +\psi _{s}(\mu ))}{ \{\frac{1}{2\pi }\int _{0}^{2\pi } \vert \psi _{s}(\mu ) +{\mathrm {e}}^{{\mathrm {i}}\beta } \vert ^{\delta } {\,\mathrm {d}}\beta \}^{\frac{1}{\delta }}} \biggl\{ \int _{0}^{2\pi } \bigl\vert \phi \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}. $$

(4.5)

Now

$$\begin{aligned} D_{\alpha }P(z)&=nP(z)+(\alpha -z)P^{\prime }(z) \\ &=nz^{s} \phi (z)+(\alpha -z) \bigl(sz^{s-1}\phi (z)+z^{s} \phi ^{\prime }(z) \bigr) \\ &=z^{s} \bigl((n-s)\phi (z)+(\alpha -z)\phi ^{\prime }(z) \bigr)+\alpha sz^{s-1} \phi (z) \\ &=z^{s}D_{\alpha }\phi (z)+s\alpha z^{s-1}\phi (z), \end{aligned}$$

which implies

$$ zD_{\alpha }P(z)=z^{s+1}D_{\alpha } \phi (z)+s\alpha P(z). $$

(4.6)

Hence, for $0\leq \theta <2\pi$, then from (4.6) we get

$$ \bigl|D_{\alpha }P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|=\bigl|{\mathrm {e}}^{{\mathrm {i}}(s+1)\theta }D_{ \alpha }\phi \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+s \alpha P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr|, $$

which gives, by using Minkowski’s inequality for $\delta \geq 1$,

$$\begin{aligned} \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha }P \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} &= \biggl\{ \int _{0}^{2\pi } \bigl\vert {\mathrm {e}}^{\mathrm {i}(s+1)\theta }D_{ \alpha }\phi \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)+s \alpha P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}} \theta \biggr\} ^{\frac{1}{\delta }} \\ & \leq \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha } \phi \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}} \theta \biggr\} ^{\frac{1}{\delta }} \\ &\quad {}+s \vert \alpha \vert \biggl\{ \int _{0}^{2 \pi } \bigl\vert P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}. \end{aligned}$$

(4.7)

Using (4.5) in (4.7) and noting that

$$ \bigl|\phi \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|=\bigl|{\mathrm {e}}^{{\mathrm {i}}s\theta }\phi \bigl({\mathrm {e}}^{ \mathrm {i}\theta }\bigr)\bigr|=\bigl|P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr)\bigr|, $$

it follows that, for every $|\alpha |\geq 1$ and $\delta \geq 1$,

$$\begin{aligned} & \biggl\{ \int _{0}^{2\pi } \bigl\vert D_{\alpha }P \bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }} \\ &\quad \leq \biggl( \frac{(n-s)( \vert \alpha \vert +\psi _{s}(\mu ))}{ \{ \frac{1}{2\pi }\int _{0}^{2\pi } \vert \psi _{s}(\mu ) +{\mathrm {e}}^{{\mathrm {i}}\beta } \vert ^{\delta } {\,\mathrm {d}}\beta \} ^{{1}/{\delta }}} +s \vert \alpha \vert \biggr) \biggl\{ \int _{0}^{2\pi } \bigl\vert P\bigl({\mathrm {e}}^{\mathrm {i}\theta }\bigr) \bigr\vert ^{\delta }{\,\mathrm {d}}\theta \biggr\} ^{\frac{1}{\delta }}, \end{aligned}$$

which is inequality (2.7), and this completes the proof of Theorem 2.9. □

5 Conclusion

We establish some Zygmund-type integral inequalities for the polar derivative of polynomials that are inspired by the classical Bernstein-type inequalities that relate the sup-norm of a polynomial to that of its derivative on the unit circle. The obtained results sharpen and generalize some already known estimates that relate the $L^{\delta }$-norms of polynomials and their polar derivatives.