Extremal problems for trinomials with fold symmetry

Abstract

The famous T. Suffridge polynomials have many extremal properties: the maximality of coefficients when the leading coefficient is maximal; the zeros of the derivative are located on the unit circle; the maximum radius of stretching the unit disk with the schlicht normalization $F(0)=0$, $F'(0)=1$; the maximum size of the unit disk contraction in the direction of the real axis for univalent polynomials with the normalization $F(0)=0$, $F(1)=1.$ However, under the standard symmetrization method $\sqrt[T]{F(z^T)}$, these polynomials go to functions, which are not polynomials. How can we construct the polynomials with fold symmetry that have properties similar to those of the Suffridge polynomial? What values will the corresponding extremal quantities take in the above-mentioned extremal problems? The paper is devoted to solving these questions for the case of the trinomials $F(z)=z+az^{1+T}+bz^{1+2T}$. Also, there are suggested hypotheses for the general case in the work.