Extremal problems for trinomials with fold symmetry

TL;DR

This paper investigates extremal properties of trinomials with fold symmetry, inspired by Suffridge polynomials, aiming to construct such polynomials with similar extremal features and explore their extremal quantities.

Contribution

It introduces a method to construct fold-symmetric trinomials with extremal properties akin to Suffridge polynomials and proposes hypotheses for the general case.

Findings

Constructed specific fold-symmetric trinomials with extremal properties.

Identified extremal quantities for these trinomials.

Proposed hypotheses for broader classes of symmetric polynomials.

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.