A new inequality for a polynomial (Q1607793)

The author proves the following: If \(p(z)=a_0+ \sum^n_{j=1} a_j z^j\) is a polynomial of degree \(n\) having no zeros in the disk \(|z|<k\) where \(k\geq 1\), then for \(1\leq t\leq n\) \[ \begin{multlined} \max_{|z|=1} \bigl |p'(z)\bigr |\leq n{1+(t/n) |a_t/a_0 |k^{t+1} \over 1+k^{t+1}+ (t/n) |a_t/a_0 |(k^{t+1}+ k^{2t})}\max_{|z|=1} \bigl|p(z) \bigr|\\ -\left\{1-{1+(t/n) |a_t/a_0 |k^{t+1} \over 1+k^{t+1}+ (t/n)|a_t/a_0 |(k^{t+1} +k^{2t})}\right\} {mn\over k^n} \end{multlined} \] This is an improvements of a result of [\textit{M. A. Qazi}, Proc. Am. Math. Soc. 115, 337-349 (1992; Zbl 0772.30006)].