Let $P(z)$ be a polynomial of degree $n$ which has no zeros in $|z|<1$, then it was proved by Liman, Mohapatra and Shah \cite{moh} that \begin{align*} &eft|zD_lpha P(z) + n\beta eft( \frac{|lpha|-1}{2}\right) P(z)\right| eq &{}\frac{n}{2}eftbrace eft|lpha +\beta eft( \frac{|lpha|-1}{2}\right) \right|+eft|z+\betaeft(\frac{|lpha|-1}{2}\right)\right|\right\rbrace \maximits_{|z|=1}|P(z)| &-\frac{n}{2}eftbrace eft|lpha + \betaeft(\frac{|lpha|-1}{2} \right) \right|-eft|z +\betaeft( \frac{|lpha|-1}{2}\right) \right|\right\rbrace \minimits_{|z|=1}|P(z)|, \end{align*} for any $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we generalize the above inequality and our result also generalizes certain well known polynomial inequalities.