Complex exponential integral means spectrums of univalent functions and Brennan conjecture

Abstract: In this paper we investigate the complex exponential integral means spectrums of univalent functions in the unit disk. We show that all integral means spectrum (IMS) functionals for complex exponents on the universal Teichmüller space, the closure of the universal Teichmüller curve, and the universal asymptotic Teichmüller space are continuous. We also show that the complex exponential integral means spectrum of any univalent function admitting a quasiconformal extension to the extended complex plane is strictly less than the universal integral means spectrum. These results extend some related ones in our recent work \cite{Jin}. Here we employ a different and more direct approach to prove the continuity of IMS functional on the universal asymptotic Teichmüller space. Additionally, we completely determine the integral means spectrums of all univalent rational functions in the unit disk. As a consequence, we show that the Brennan conjecture is true for this class of univalent functions. Finally, we present some remarks and raise some problems and conjectures regarding IMS functionals on Teichmüller spaces, univalent rational functions, and a multiplier operator whose norm is closely related to the Brennan conjecture.