Extremal problems for typically real odd polynomials (Q6607753)

The paper deals with the properties of odd polynomials with real coefficients. Main attention is paid so called typical real polynomials \(F(z) = z + \sum\limits_{j=2}^{N} a_j z^{2j-1}\), i.e. those which are real when \(z\) is real and satisfying inequality \[ {{\mathrm{Im}}} F(z)\cdot {{\mathrm{Im}}} z > 0 \] for the remaining points in the unit disc.\par A sharp bounds for the coefficient of the third power in the normalized typically real odd polynomials are found. The corresponding extremizers are constructed.