ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971880

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

摘要:

Summary1984 has been an exciting year for complex analysis. It even brought strong rumors that the Riemann hypothesis had been proved, but so far, the rumor has not been confirmed. However, we know for sure that the difficult Bieberbach conjecture has been settled this year. As many of you know, this famous conjecture of 1916 concerns the classSof normalized injective holomorphic functions. That class consists of the 1–1 holomorphic functions from the unit discUinto the complex plane C with a power series of the formf(z)=z+ a2z2+…+anzn+…,|z|<1.The conjecture asserts that |an|⩽nfor veryfinSand everyn. Louis de Branges has proved this conjecture as well as some stronger conjectures for the classS.Each of the following items has played an essential role in the proof:(i)Löwner's partial differential equation for so-called Löwner chains {f,(z)} of injective holomorphic functions fromUto C.(ii)The observations of Lebedev and I. M. Milin, especially their inspired conjecture for the so-called logarithmic coefficients offinS, that is, the coefficients in the expansionfor a branch of log {f(z)/z}.(iii)De Branges' striking breakthrough, namely, the creation of a functional associated with the Lebedev-Milin conjecture which varies monotonically along Löwner chains.(iv)De Branges' introduction and solution of a system of differential equations which he devised to make the functional manageable.(v)A positivity result for hypergeometric functions which is a tool in establishing the monotonicity of the functional.Of the above, (i) dates back to 1923, while (ii) and (v) are relatively recent. The Lebedev-Milin observations date from the years 1965–1970 and became well known in the West only around 1977. The hypergeometric functions result occurs in work of Askey and Gasper of 1976.

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971872

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971873

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971874

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971875

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971876

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971877

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971878

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971879

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor

ISSN:0002-9890    

DOI:10.1080/00029890.1986.11971881

出版商:Taylor&Francis    

年代:1986

数据来源: Taylor