Distortion theorems for polynomials on a circle (Q2736053)

The author considers polynomials NEWLINE\[NEWLINEP(z)= \sum^n_{k=0} c_kz^k, \quad c_n\neq 0,NEWLINE\]NEWLINE for points \(z=e^{i\varphi}\) on the unit circle. He proves new and interesting estimates for the derivatives NEWLINE\[NEWLINE{\partial\text{Re} P(e^{i \varphi}) \over\partial \varphi},\;{\partial\bigl|P(e^{i\varphi}) \bigr |^2 \over\partial\varphi} \text{ and } {\partial\arg \bigl(P(e^{i\varphi}) \bigr)\over\partial \varphi}NEWLINE\]NEWLINE using the moduli of \(c_0\) and \(c_n\) and the minimum and the maximum of the real part and of the modulus of \(P\) on the unit circle. Since most of these inequalities are to involved to be cited here, we give only two examples where the estimates are short:NEWLINENEWLINENEWLINE(1) If \(P\) is non-vanishing in the open unit disc, then NEWLINE\[NEWLINE{\partial\arg \bigl(P(e^{i\varphi}) \bigr)\over\partial \varphi}\leq{n-1\over 2}+\sqrt{{|c_n|\over 4|c_0|}}.NEWLINE\]NEWLINE (2) If all zeroes of \(P\) lie in the closed unit disc, then NEWLINE\[NEWLINE{\partial\arg \bigl(P(e^{i\varphi}) \bigr)\over\partial\varphi} \geq{n+1\over 2}-\sqrt{ {|c_0|\over 4|c_n|}}.NEWLINE\]