A generalization of Ankeny and Rivlin’s result to $(s-1)^{st}$ derivative of a polynomial

Abstract

For an arbitrary entire function f(z) let $M(f,r) = max_{|z|=r}|f(z)|, (r > 0)$ and $\Vert f\Vert = max_{|z|=1}|f(z)|$. For a polynomial $p(z) = a_0 + \sum _{j=t}^{n}a_jz^j, (1 \le t \le n)$, of degree n having no zeros in $|z| < k, (k\ge 1)$, with $m = min_{|z|=k}|p(z)|$, it is known that for $R \ge 1$

$$\begin{aligned} M(p,R)\le & {} \frac{R^n + k^t}{1 + k^t}\Vert p\Vert - \frac{R^n-1}{1+k^t}m - \frac{n}{1+k^t}\left( \frac{(\Vert p\Vert - m)^2 - (1+k^t)^2|a_n|^2}{(\Vert p\Vert -m)} \right) \\&\times \left\{ \frac{(R-1)(\Vert p\Vert -m)}{(\Vert p\Vert -m)+(1+k^t)|a_n|} - \ln \left( 1 + \frac{(R-1)(\Vert p\Vert -m)}{(\Vert p\Vert -m)+(1+k^t)|a_n|}\right) \right\} \end{aligned}$$

and we have obtained for $R \ge 1$ and $1 \le s \le n$

$$\begin{aligned} M(p^{(s-1)},R)\le & {} \frac{R^{n-s+1}-1}{n-\overline{s-1}}.\frac{n(n-1)\ldots (n-\overline{s-1})}{1+k^t} (\Vert p\Vert -m) \\&-\frac{n(n-1)\ldots (n-\overline{s-1})}{1+k^t} \left( \frac{(\Vert p\Vert -m)^2 - (1+k^t)^2 |a_n|^2}{\Vert p\Vert -m}\right) \\&\times \left\{ \frac{(R-1)(\Vert p\Vert -m)}{(\Vert p\Vert -m)+(1+k^t)|a_n|} - \ln \left( 1 + \frac{(R-1)(\Vert p\Vert -m)}{(\Vert p\Vert -m)+(1+k^t)|a_n|}\right) \right\} + \Vert p^{(s-1)}\Vert , \end{aligned}$$

thereby suggesting a generalization of Ankeny and Rivlin’s result

$$\begin{aligned} M(P,R) \le \frac{1+R^n}{2}, \left( P(z): \text { a polynomial of degree } n \text { having no zeros in } |z| < 1, \text { with } \Vert p\Vert =1\right) \end{aligned}$$

to $(s-1)^{st}$ derivative of a polynomial, as well as generalization of known result to $(s-1)^{st}$ derivative of a polynomial. Certin associated results have also been discussed.