單葉亞純螺旋象函數(shù)的刻畫和積分表示

錢繼曉

單葉亞純螺旋象函數(shù)的刻畫和積分表示

錢繼曉

（南京理工大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，江蘇 南京 210094）

黎曼映射定理為復(fù)變函數(shù)的性質(zhì)提供了幾何刻畫；Carathéodory收斂定理把函數(shù)像域的收斂與函數(shù)的收斂性緊密聯(lián)系起來(lái)。利用黎曼映射定理、極值原理和Carathéodory收斂定理，研究極點(diǎn)在原點(diǎn)和極點(diǎn)在點(diǎn) (0<<1)的單葉亞純螺旋象函數(shù)，得到了相應(yīng)函數(shù)族的解析刻畫和積分表示。

單葉函數(shù)；亞純函數(shù)；螺旋象函數(shù)

1 引言

2 單葉亞純螺旋象函數(shù)的刻畫

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[1] BIEBERBACH L. über einige extremal probleme im Gebiete der konformen abbildung[J]. Mathematische annalen, 1916, 77(2): 153–172.

[2] LOWNER K. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises[J]. Mathematische annalen, 1923, 89(1): 103–121.

[3] GARABEDIAN P, SCHIFFER M. A proof of the Bieberbach conjecture for the fourth coefficient[J]. Journal of rational mechanics and analysis, 1955, 4: 427–465.

[4] PEDERSON R N. A proof of the Bieberbach conjecture for the sixth coefficient[M]. Carnegie institute of technology, department of mathematics, 1968.

[5] OZAWA M. An elementary proof of local maximality for a6[C]//Kodai Mathematical seminar reports, department of mathematics, Tokyo institute of technology, 1968, 20(4): 437–439.

[6] PEDERSON R N, SCHIFFERR M. A proof of the Bieberbach conjecture for the fifth coefficient[J]. Archive for rational mechanics and analysis, 1972, 45(3): 161–193.

[7] BRANGES L D, A proof of the Bieberbach conjecture[J]. Acta mathematica, 1985, 154(1): 137–152.

[8] PFALTZGRAFF J A, PINCHUK B. A variational method for classes of meromorphic functions[J]. J. Analyse math, 1971, 24: 101–150.

[9] OHNO R. Characterizations for concave functions and integral representations[J]. Topics in finite or infinite dimensional complex analysis, 2013: 203–216.

[10] POMMERENKE C. Boundary behaviour of conformal maps[M]. Springer science & business media, 2013.

Characterization and Integral Representation of Univalent Metamorphic Spirallike Functions

QIAN Ji-xiao

(School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing Jiangsu 210094, China)

The Riemann mapping theorem provides a geometric characterization for the properties of complex functions; the Carathéodory convergence theorem closely links the convergence of the function image field with the convergence of the function. Using the Riemann mapping theorem, the extreme value principle and the Carathéodory convergence theorem, the univalent meromorphic spirallike functions with the pole at the origin and the pole at the p point (0

univalent functions; meromorphic functions; spirallike functions

2022-03-20

江蘇省研究生科研與實(shí)踐創(chuàng)新計(jì)劃項(xiàng)目 (KYCX21–0247)

錢繼曉（1983—），男，江蘇連云港人，碩士研究生，研究方向：復(fù)分析。

O174.52

A

2095-9249（2022）03-0011-05

〔責(zé)任編校：吳侃民〕