Let *P*(*z*) be a polynomial of degree *n* which does not vanish in
$$|z|<1$$
. Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that
$$\begin{aligned} \Bigg |z^s P^{(s)}+\beta \dfrac{n_s}{2^s}P(z)\Bigg |\le \dfrac{n_s}{2}\Bigg (\bigg |1+\dfrac{\beta }{2^s}\bigg |+\bigg | \dfrac{\beta }{2^s}\bigg |\Bigg )\underset{|z|=1}{\max }|P(z)|, \end{aligned}$$
for every
$$\beta \in \mathbb C$$
with
$$|\beta |\le 1,1\le s\le n$$
and
$$|z|=1.$$

The
$$L^{\gamma }$$
analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved
$$\begin{aligned}&\Bigg \{\int _0^{2\pi }\Big |e^{is\theta }P^{(s)}(e^{i\theta })+\beta \dfrac{n_s}{2^s}P(e^{i\theta })\Big |^ {\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }\\&\quad \le n_s\Bigg \{\int _0^{2\pi }\Big |\Big (1+\dfrac{\beta }{2^s}\Big )e^{i\alpha }+\dfrac{\beta }{2^s}\Big |^{\gamma } \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }\dfrac{\Bigg \{\int _0^{2\pi }\big |P(e^{i\theta })\big |^{\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }}{\Bigg \{\int _{0}^{2\pi }\big |1+e^{i\alpha }\big |^\gamma \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }}, \end{aligned}$$
where
$$n_s=n(n-1)\ldots (n-s+1)$$
and
$$0\le \gamma <\infty $$
.

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