A Zygmund-type integral inequality for polynomials

Abstract

Let P(z) be a polynomial of degree n which does not vanish in $$|z|<1$$ | z | < 1 . Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that $$\begin{aligned} \Bigg |z^s P^{(s)}+\beta \dfrac{n_s}{2^s}P(z)\Bigg |\le \dfrac{n_s}{2}\Bigg (\bigg |1+\dfrac{\beta }{2^s}\bigg |+\bigg | \dfrac{\beta }{2^s}\bigg |\Bigg )\underset{|z|=1}{\max }|P(z)|, \end{aligned}$$ | z s P ( s ) + β n s 2 s P ( z ) | ≤ n s 2 ( | 1 + β 2 s | + | β 2 s | ) max | z | = 1 | P ( z ) | , for every $$\beta \in \mathbb C$$ β ∈ C with $$|\beta |\le 1,1\le s\le n$$ | β | ≤ 1 , 1 ≤ s ≤ n and $$|z|=1.$$ | z | = 1 . The $$L^{\gamma }$$ L γ analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved $$\begin{aligned}&\Bigg \{\int _0^{2\pi }\Big |e^{is\theta }P^{(s)}(e^{i\theta })+\beta \dfrac{n_s}{2^s}P(e^{i\theta })\Big |^ {\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }\\&\quad \le n_s\Bigg \{\int _0^{2\pi }\Big |\Big (1+\dfrac{\beta }{2^s}\Big )e^{i\alpha }+\dfrac{\beta }{2^s}\Big |^{\gamma } \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }\dfrac{\Bigg \{\int _0^{2\pi }\big |P(e^{i\theta })\big |^{\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }}{\Bigg \{\int _{0}^{2\pi }\big |1+e^{i\alpha }\big |^\gamma \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }}, \end{aligned}$$ { ∫ 0 2 π | e i s θ P ( s ) ( e i θ ) + β n s 2 s P ( e i θ ) | γ d θ } 1 γ ≤ n s { ∫ 0 2 π | ( 1 + β 2 s ) e i α + β 2 s | γ d α } 1 γ { ∫ 0 2 π | P ( e i θ ) | γ d θ } 1 γ { ∫ 0 2 π | 1 + e i α | γ d α } 1 γ , where $$n_s=n(n-1)\ldots (n-s+1)$$ n s = n ( n - 1 ) … ( n - s + 1 ) and $$0\le \gamma <\infty $$ 0 ≤ γ < ∞ . In this paper, we generalize this and some other related results.