V. K. Jain: On the Derivatives of a Polynomial. p. 339-347

Abstract:

For a polynomial $p(z)$ of degree $n$, having all its zeros in $\vert z\vert \leq k, ( k \geq 1)$, we obtain a refinement of a known result [2]

$$\nonumber \max_{\vert z\vert=1} \vert p'(z)\vert \geq \frac{n}{1+k^n} \max_{\vert z\vert=1}\vert p(z)\vert,$$

(by using certain coefficients of $p(z)$), and an inequality, similar to known result involving $s^{th}$ derivative, ( $2 \leq s \leq n$), instead of the first derivative of $p(z)$ (and better than the similar inequality, obtained by repeated applications of known result, in many cases).

Key Words: polynomial, derivatives, zeros in $\vert z\vert \leq k, k \geq 1$, certain coefficients, generalization of Schwarz's lemma.

2010 Mathematics Subject Classification: Primary 30C10,
Secondary 30A10.