TY - JOUR

T1 - Sharp inequalities for logarithmic coefficients and their applications

AU - Ponnusamy, Saminathan

AU - Sugawa, Toshiyuki

N1 - Funding Information:
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) of DST , India ( MTR/2017/000367 ).

PY - 2021/2

Y1 - 2021/2

N2 - 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.

AB - 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.

KW - De Branges theorem

KW - Logarithmic coefficient

KW - Milin conjecture

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U2 - 10.1016/j.bulsci.2020.102931

DO - 10.1016/j.bulsci.2020.102931

M3 - Article

AN - SCOPUS:85097080942

VL - 166

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

SN - 0007-4497

M1 - 102931

ER -