On the derivative of a self-reciprocal polynomial

Similar to a polynomial P(z) with zeros w(1),w(2), . . .,w(n), being called a self-inversive polynomial (see [8]) if {w(1),w(2), . . .,w(n)} = {1/(w(1)) over bar, 1/(w(2)) over bar, . . ., 1/(w(n)) over bar}, we have called a polynomial p(z) with zeros z(1), z(2), . . ., z(n), a self-reciprocal polynomial if {z(1), z(2), . . ., z(n)} = {1/z(1), 1/z(2), . . ., 1/z(n)} and have obtained for a self-reciprocal polynomial p(z) of degree n (vertical bar z vertical bar=1)max vertical bar p'(z)vertical bar + vertical bar p'((z) over bar)vertical bar) = n (vertical bar z vertical bar=1)max vertical bar p(z)vertical bar, similar to (vertical bar z vertical bar=1)max vertical bar p'(z)vertical bar = n/2 (vertical bar z vertical bar=1)max vertical bar p(z)vertical bar, for a self-inversive polynomial P(z) of degree n ([8]).