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  • 1
    Publication Date: 2019
    Description: 〈h3〉Abstract〈/h3〉 〈p〉Let 〈em〉P〈/em〉(〈em〉z〈/em〉) be a polynomial of degree 〈em〉n〈/em〉 which does not vanish in 〈span〉 〈span〉\(|z|〈1\)〈/span〉 〈/span〉. Then it was proved by Hans and Lal (Anal Math 40:105–115, 〈span〉2014〈/span〉) that 〈span〉 〈span〉$$\begin{aligned} \Bigg |z^s P^{(s)}+\beta \dfrac{n_s}{2^s}P(z)\Bigg |\le \dfrac{n_s}{2}\Bigg (\bigg |1+\dfrac{\beta }{2^s}\bigg |+\bigg | \dfrac{\beta }{2^s}\bigg |\Bigg )\underset{|z|=1}{\max }|P(z)|, \end{aligned}$$〈/span〉 〈/span〉for every 〈span〉 〈span〉\(\beta \in \mathbb C\)〈/span〉 〈/span〉 with 〈span〉 〈span〉\(|\beta |\le 1,1\le s\le n\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(|z|=1.\)〈/span〉 〈/span〉〈/p〉 〈p〉The 〈span〉 〈span〉\(L^{\gamma }\)〈/span〉 〈/span〉 analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 〈span〉2016〈/span〉) who under the same hypothesis proved 〈span〉 〈span〉$$\begin{aligned}&\Bigg \{\int _0^{2\pi }\Big |e^{is\theta }P^{(s)}(e^{i\theta })+\beta \dfrac{n_s}{2^s}P(e^{i\theta })\Big |^ {\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }\\&\quad \le n_s\Bigg \{\int _0^{2\pi }\Big |\Big (1+\dfrac{\beta }{2^s}\Big )e^{i\alpha }+\dfrac{\beta }{2^s}\Big |^{\gamma } \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }\dfrac{\Bigg \{\int _0^{2\pi }\big |P(e^{i\theta })\big |^{\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }}{\Bigg \{\int _{0}^{2\pi }\big |1+e^{i\alpha }\big |^\gamma \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }}, \end{aligned}$$〈/span〉 〈/span〉where 〈span〉 〈span〉\(n_s=n(n-1)\ldots (n-s+1)\)〈/span〉 〈/span〉 and 〈span〉 〈span〉\(0\le \gamma 〈\infty \)〈/span〉 〈/span〉.〈/p〉 〈p〉In this paper, we generalize this and some other related results.〈/p〉
    Print ISSN: 2193-5343
    Electronic ISSN: 2193-5351
    Topics: Mathematics
    Published by Springer
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