Narrow Results

This paper considers the well-known Erdös–Lax and Turán-type inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. The obtained results produce inequalities that are sharper than the previous ones. Moreover, a numerical example is presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.

In this paper, some generalizations of lower bound estimates for the maximum modulus of the $t$th polar derivative $D_{{\xi }_t}{D}_{{\xi }_{t-1}}\dots D_{{\xi }_2}{D}_{{\xi }_1}$, where ${\xi }_t\in ℂ$, $1\le t<n$ of polynomials are established under the assumption that one of the underlying polynomials $p$ and $q$ does not vanish in $\left|z\right|>k,k\le 1$. The obtained estimates include several known lower bounds for polar derivatives as special cases.