Some inequalities for self-reciprocal polynomials (Q1815057)

A polynomial \(P(z)=a_nz^n+a_{n-1} z^{n-1} + \cdots+a_0\) of degree \(n\), is said to be a ``self-reciprocal'' polynomial of degree \(n\), if it satisfies \[ P(z)=z^nP(1/z). \] In this paper the following result is proved. If \[ P(z)=\sum^n_{j=0} (a_j+ib_j)z^j=P_1(z) + iP_2(z), \quad \text{where} \quad P_1(z) = \sum^n_{j=0} a_jz^j,\;P_2(z) = \sum^n_{j=0} b_jz^j \] are ``self-reciprocal'' polynomials of degree \(n\), then \[ \max_{|z|=1} \bigl|P_1'(z) \pm P_2'(z) \bigr|\leq {n\over 2} \max_{|z |=1} \bigl|P(z) \bigr|. \] The extremal polynomial have the form \(p(z)=z^{2m} + 2iz^m+1\), \(n=2m\).