Sharp inequalities for logarithmic coefficients and their applications

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Abstract

I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Milin conjecture. Recently, O. Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roth's ideas, we will show more general sharp inequalities with convex sequences as weight functions. By specializing the sequence, we can obtain an abundant number of sharp inequalities on logarithmic coefficients, some of which are provided in Appendix. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results.

Section snippets

Estimates of logarithmic coefficients

Let A denote the set of normalized analytic functions on the open unit disk D={z∈C:|z|<1} and S denote its subclass of univalent functions. We define the logarithmic coefficients of f∈S by the formulalog⁡f(z)z=2∑n=1∞γnzn. Throughout the discussion, γn:=γn(f) denote the logarithmic coefficients of a function f∈S. Louis de Branges [5] solved the long-standing Bieberbach conjecture by showing the Milin conjecture (see also [8]): For each n≥1,∑k=1nk(n−k+1)|γk|2≤∑k=1nn−k+1k, where equality holds if

Computer-assisted proof of the inequality for non-convex sequences

In the previous section, we presented an inequality of the logarithmic coefficients γn of a univalent function f∈S for a convex sequence pn. The inequality may hold even if pn is not convex; namely, some of λn=pn−2pn+1+pn+2 are negative. We shall review the idea due to Roth [16] and then reformulate it in a convenient form so that one can check the conditions by using computers.

We recall the proof of the Milin conjecture (1.2) by following FitzGerald and Pommerenke [8]. Fix n≥1. The key idea is

Applications

Our next result is related to a transform hf of f∈S introduced by Danikas and Ruscheweyh [4]:hf(z):=∫0ztf′(t)f(t)dt. It was conjectured in [4] that the transform hf belongs to S for each f∈S. This conjecture remains open. Roth [16] applied his inequality (2.5) to obtain the sharp H2 norm estimate of hf for f∈S. We now introduce the classU={f∈A:|Uf(z)|<1 for z∈D}, whereUf(z)=f′(z)(zf(z))2−1,z∈D. It is known that U⊂S. See [3] and also [9], [12], [13] and the references therein. We will say that f∈

Declaration of Competing Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgements

The authors thank the anonymous referee for careful checking and suggestions to improve the exposition. Especially, they learnt from the referee that de Branges' paper [6] contains a more general result than our Theorem 1.1 (but with only a sketchy proof as was mentioned in Introduction). The present research was supported by JSPS Grant-in-Aid for Scientific Research (B) 22340025 and JP17H02847. The work of the first author is supported by Mathematical Research Impact Centric Support (MATRICS)

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