A Zygmund-type integral inequality for polynomials

Let P(z) be a polynomial of degree n which does not vanish in |z|<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|<1$$\end{document}. Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that |zsP(s)+βns2sP(z)|≤ns2(|1+β2s|+|β2s|)max|z|=1|P(z)|,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Bigg |z^s P^{(s)}+\beta \dfrac{n_s}{2^s}P(z)\Bigg |\le \dfrac{n_s}{2}\Bigg (\bigg |1+\dfrac{\beta }{2^s}\bigg |+\bigg | \dfrac{\beta }{2^s}\bigg |\Bigg )\underset{|z|=1}{\max }|P(z)|, \end{aligned}$$\end{document}for every β∈C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in \mathbb C$$\end{document} with |β|≤1,1≤s≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\beta |\le 1,1\le s\le n$$\end{document} and |z|=1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|=1.$$\end{document} The Lγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\gamma }$$\end{document} analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved {∫02π|eisθP(s)(eiθ)+βns2sP(eiθ)|γdθ}1γ≤ns{∫02π|(1+β2s)eiα+β2s|γdα}1γ{∫02π|P(eiθ)|γdθ}1γ{∫02π|1+eiα|γdα}1γ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Bigg \{\int _0^{2\pi }\Big |e^{is\theta }P^{(s)}(e^{i\theta })+\beta \dfrac{n_s}{2^s}P(e^{i\theta })\Big |^ {\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }\\&\quad \le n_s\Bigg \{\int _0^{2\pi }\Big |\Big (1+\dfrac{\beta }{2^s}\Big )e^{i\alpha }+\dfrac{\beta }{2^s}\Big |^{\gamma } \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }\dfrac{\Bigg \{\int _0^{2\pi }\big |P(e^{i\theta })\big |^{\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }}{\Bigg \{\int _{0}^{2\pi }\big |1+e^{i\alpha }\big |^\gamma \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }}, \end{aligned}$$\end{document}where ns=n(n-1)…(n-s+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_s=n(n-1)\ldots (n-s+1)$$\end{document} and 0≤γ<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \gamma <\infty $$\end{document}. In this paper, we generalize this and some other related results.

In this paper, we generalize this and some other related results.

Introduction
Let P n be the class of polynomials P(z) = n v=0 a v z v of degree n and P (s) (z) be its sth derivative. For P ∈ P n , we have (1. 2) The inequality (1.1) is a classical result of Bernstein [10], whereas the inequality (1.2) is due to Zygmund [13] who proved it for all trigonometric polynomials of degree n and not only for those of the form P(e iθ ). Arestov [1] proved that (1.2) remains true for 0 < γ < 1 as well. If we let γ → ∞ in (1.2), we get (1.1). The above two inequalities (1.1) and (1.2) can be sharpened, if we restrict ourselves to the class of polynomials having no zeros in |z| < 1. In fact, if P ∈ P n and P(z) = 0 in |z| < 1, then (1.1) and (1.2) can be, respectively, replaced by The inequality (1.3) was conjectured by Erdös and later proved by Lax [9], whereas (1.4) was proved by De-Bruijn [4] for γ ≥ 1. Further, Rahman and Schmeisser [11] have shown that (1.4) holds for 0 < γ < 1 as well. If we let γ → ∞ in inequality (1.4), we get (1.3).
Recently, Hans and Lal [6] generalized (1.6) and (1.7) for the sth derivative of polynomials and proved the following results.
Theorem C If P ∈ P n and P(z) = 0 in |z| < 1, then for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and The result is best possible and equality in (1.10) holds for P(z) = az n + b with |a| = |b| = 1.

Main results
The main aim of this paper is to prove an L γ analog of Theorem B and thereby to obtain a generalization of Theorem C. More precisely, we prove where here and throughout m = min |z|=1 |P(z)| and E γ is defined by (1.11).
The result is best possible and equality in (2.1) holds for P(z) = az n + b with |a| = |b| = 1. Now, we present and discuss some consequences of this result. First, we point out that inequalities involving polynomials in the Chebyshev norm on the unit circle in the complex plane are a special case of the polynomial inequalities involving the integral norm. For example, if we let γ → ∞ in (2.1) and choose the argument of δ suitably with |δ| = 1, we get (1.9).

Remark 2.2
For δ = 0, Theorem 2.1 reduces to Theorem C. If we take s = 1 in (2.1), we get the following result which provides an L γ analogue of (1.7).
Remark 2.4 Inequality (1.7) can be obtained by letting γ → ∞ and by choosing the argument of δ suitably with |δ| = 1 in (2.2). Several other interesting results easily follow from Theorem 2.1. Here, we mention a few of these. Taking β = 0 in (2.1), we immediately get the following result.
For s = 1 and δ = 0, inequality (2.3) reduces to inequality (1.4). The following corollary which is a refinement as well as a generalization of (1.3) is obtained by letting γ → ∞ and by choosing the argument of δ with |δ| = 1 suitably in (2.3).

Corollary 2.6
If P ∈ P n and P(z) = 0 in |z| < 1, then for 1 ≤ s ≤ n , we have Remark 2.7 For s = 1, Corollary 2.6 reduces to a result of Aziz and Dawood [2]. For the proof of Theorem 2.1, we need the following lemmas.

Lemmas
Lemma 3.1 Let F ∈ P n and F(z) has all its zeros in |z| ≤ 1. If P(z) is a polynomial of degree at most n such that then for any β ∈ C with |β| ≤ 1 and 1 ≤ s ≤ n, The above lemma is due to Hans and Lal [6].
By applying Lemma 3.1 to polynomials P(z) and z n min |z|=1 |P(z)|, we get the following result.
The following lemma is due to Aziz and Shah [3]. A, B, C are non-negative real numbers such that B + C ≤ A, then for every real number α,