On univalent polynomials with critical points on the unit circle

Abstract

Brannan showed that a normalized univalent polynomial of the form (P(z)=z+a2 z2+...+ an-1zn-1+znn) is starlike if and only if (a2=...=an-1=0). We give a new and simple proof of his result, showing further that it is also equivalent to the membership of P in the Noshiro–Warschawski class of univalent functions whose derivative has positive real part in the disk. Both proofs are based on the Fejér lemma for trigonometric polynomials with positive real part.

Keywords:

• Polynomials
• critical points
• starlike functions
• functions with positive real part
• Fejér lemma

AMS Subject Classifications:

• 30C10
• 30C45

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Secretaría de Estado de Investigación, Desarrollo e Innovación, MINECO and FEDER/ERDF [MTM2015-65792-P], [MTM2015-69323-REDT].