Applications of Livingston-type inequalities to the generalized Zalcman functional

This is the peer-reviewed version of the following article: Efraimidis, I., & Vukotić, D. Applications of Livingston‐type inequalities to the generalized Zalcman functional. Mathematische Nachrichten 291.10 (2018): 1502-1513, which has been published in final form at https://doi.org/10.1002/mana.201700022. This article may be used for non-commercial purposes in accordance with Wiley-VCH Terms and Conditions for Self-ArchivingWe obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro–Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [6]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [17]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the problemThe authors are partially supported by MTM2015-65792-P from MINECO/FEDER-EU and the Thematic Research Network MTM2015-69323-REDT, MINECO, Spai

On univalent polynomials with critical points on the unit circle

This is an Accepted Manuscript of an article published by Taylor & Francis in Complex Variables and Elliptic Equations on 03 Jan 2018, available online: http://www.tandfonline.com/10.1080/17476933.2017.1420065Brannan 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 partThis work was supported by Secretaría de Estado de Investigación, Desarrollo e Innovación, MINECO and FEDER/ERDF [MTM2015-65792-P], [MTM2015-69323-REDT

Superposition operators and the order and type of entire functions

We distinguish between entire functions of diffent order or type by the behavior of their associated superposition operators on subsets of Besov spaces or the little Bloch space

Univalent interpolation on Besov spaces and superposition into Bergman spaces

We characterize the superposition operators from an analytic Besov space or the little Bloch space into a Bergman space in terms of the order and type of the symbol. We also determine when these operators are continuous or bounded. Along the way, we prove new non-centered Trudinger-Moser inequalities and solve the problem of interpolation by univalent functions in analytic Besov spaces