﻿ 一类单叶接近凸调和映射

# 一类单叶接近凸调和映射A Class of Univalent Close-to-Convex Harmonic Mapping

Abstract: This paper investigates a class of univalent close-to-convex harmonic mappings, which is the gener-alization of analytic functions whose derivatives have positive real parts. We discuss the following properties of functions in this class: deviation theorem, the radius of convexity, close-to-convexity of partial sums, extremal functions, and we also study the subclass of functions with initial zero coef-ficients.

1. 预备知识

${H}_{0}$ 是单位圆盘 $D=\left\{z\in ℂ:|z|<1\right\}$ 上的复值调和映射 $f=h+\stackrel{¯}{g}$ 组成的函数类，其中h和g是D上的解析函数，分别称为f的解析部分和反解析部分， $h\left(0\right)=g\left(0\right)=0$${h}^{\prime }\left(0\right)=1$。h和g有级数表示

$h\left(z\right)=z+\underset{n=2}{\overset{\infty }{\sum }}{a}_{n}{z}^{n},g\left(z\right)=\underset{n=1}{\overset{\infty }{\sum }}{b}_{n}{z}^{n},z\in D.$

$\Re =\left\{f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n}:对某一个{\theta }_{0}\in \left[0,2\text{π}\right],\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{f}^{\prime }\left(z\right)>0,z\in D\right\}.$

$H:=\left\{f=h+\stackrel{¯}{g}\in {H}_{0}:对某个{\theta }_{0}\in \left[0,2\text{π}\right],\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{h}^{\prime }\left(z\right)>|{g}^{\prime }\left(z\right)|,z\in D\right\}.$

2. 偏差定理

$|{a}_{n}|+|{b}_{n}|\le \frac{2}{n}\mathrm{cos}{\theta }_{0},n=2,3,\cdots ,$

$|{h}^{\prime }\left(z\right)|+|{g}^{\prime }\left(z\right)|\le \mathrm{cos}{\theta }_{0}\frac{1+|z|}{1-|z|}+|\mathrm{sin}{\theta }_{0}|,$

$\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{h}^{\prime }\left(z\right)\ge \mathrm{cos}{\theta }_{0}\frac{1-|z|}{1+|z|}+|{g}^{\prime }\left(z\right)|,$

$||h\left(z\right)|-|g\left(z\right)||\le |h\left(z\right)|+|g\left(z\right)|\le \mathrm{cos}{\theta }_{0}\left[-|z|-2\mathrm{log}\left(1-|z|\right)\right]+|\mathrm{sin}{\theta }_{0}||z|,$

$||h\left(z\right)|-|g\left(z\right)||\ge \mathrm{cos}{\theta }_{0}\left[-|z|+2\mathrm{log}\left(1+|z|\right)\right].$ (1)

$|{a}_{n}+{\text{e}}^{i\left(\theta -{\theta }_{0}\right)}{b}_{n}|\le \frac{2}{n}\mathrm{cos}{\theta }_{0},n=2,3,\cdots ,$

$|{h}^{\prime }\left(z\right)+{\text{e}}^{i\left(\theta -{\theta }_{0}\right)}{g}^{\prime }\left(z\right)|\le \mathrm{cos}{\theta }_{0}\frac{1+|z|}{1-|z|}+|\mathrm{sin}{\theta }_{0}|,$

$\mathrm{Re}{\text{e}}^{i{\theta }_{0}}\left({h}^{\prime }\left(z\right)+{\text{e}}^{i\left(\theta -{\theta }_{0}\right)}{g}^{\prime }\left(z\right)\right)>\mathrm{cos}{\theta }_{0}\frac{1-|z|}{1+|z|},$

$|h\left(z\right)+{\text{e}}^{i\left(\theta -{\theta }_{0}\right)}g\left(z\right)|\le \mathrm{cos}{\theta }_{0}\left[-|z|-2\mathrm{log}\left(1-|z|\right)\right]+|\mathrm{sin}{\theta }_{0}||z|,$

$|h\left(z\right)+{\text{e}}^{i\left(\theta -{\theta }_{0}\right)}g\left(z\right)|\ge \mathrm{cos}{\theta }_{0}\left[-|z|+2\mathrm{log}\left(1+|z|\right)\right].$

$|{a}_{n}|+|{b}_{n}|\le \frac{2}{n}\mathrm{cos}{\theta }_{0},n=2,3,\cdots ,$

$|{h}^{\prime }\left(z\right)|+|{g}^{\prime }\left(z\right)|\le \mathrm{cos}{\theta }_{0}\frac{1+|z|}{1-|z|}+|\mathrm{sin}{\theta }_{0}|,$

$\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{h}^{\prime }\left(z\right)\ge \mathrm{cos}{\theta }_{0}\frac{1-|z|}{1+|z|}+|{g}^{\prime }\left(z\right)|,$

$||h\left(z\right)|-|g\left(z\right)||\le |h\left(z\right)|+|g\left(z\right)|\le \mathrm{cos}{\theta }_{0}\left[-|z|-2\mathrm{log}\left(1-|z|\right)\right]+|\mathrm{sin}{\theta }_{0}||z|,$

$||h\left(z\right)|-|g\left(z\right)||\ge \mathrm{cos}{\theta }_{0}\left[-|z|+2\mathrm{log}\left(1-|z|\right)\right].$

3. 凸半径

$z{{f}^{″}}_{0}\left(z\right)/{{f}^{″}}_{0}\left(z\right)+1=\left(1+2z-{z}^{2}\right)/\left(1-{z}^{2}\right)$

4. 部分和的接近凸性

${f}_{n}\left(z\right)={h}_{n}\left(z\right)+\stackrel{¯}{{g}_{n}\left(z\right)}=z+{\sum }_{k=2}^{n}{a}_{k}{z}^{k}+{\sum }_{k=2}^{n}\stackrel{¯}{{b}_{k}}{\stackrel{¯}{z}}^{k}$，那么 ${f}_{n}\left(z\right)$$|z|<\frac{1}{2}$ 中是接近凸的，其中 $n=2,3,\cdots$。这个结果是强的。

${f}_{\theta }\left(z\right)=h\left(z\right)+{\text{e}}^{i\left(\theta -{\theta }_{0}\right)}g\left(z\right)\in \Re$

$\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{{h}^{\prime }}_{n}\left(z\right)>|{{g}^{\prime }}_{n}\left(z\right)|,$

5. 极值性质

${\theta }_{0}=\left[0,2\text{π}\right]$${H}_{{\theta }_{0}}=\left\{f\in H:\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{h}^{\prime }\left(z\right)>|{g}^{\prime }\left(z\right)|\right\}$

$\begin{array}{c}{A}_{r}\left(f\right)={\iint }_{D}{J}_{f}\left(z\right)\text{d}x\text{d}y={\iint }_{D}\left({|{h}^{\prime }\left(z\right)|}^{2}-{|{g}^{\prime }\left(z\right)|}^{2}\right)\text{d}x\text{d}y\\ \le {\iint }_{D}{|{h}^{\prime }\left(z\right)|}^{2}\text{d}x\text{d}y={A}_{r}\left(h\right)\le {A}_{r}\left({f}_{{\theta }_{0}}\right)\le {A}_{r}\left({f}_{0}\right).\end{array}$

6. 幂级数展开式中除首项外前有限项系数为零的调和映射

$|{h}^{\prime }\left(z\right)|+|{g}^{\prime }\left(z\right)|\le \mathrm{cos}{\theta }_{0}\frac{1+{|z|}^{k-1}}{1-{|z|}^{k-1}}+|\mathrm{sin}{\theta }_{0}|,$

$\mathrm{Re}{\text{e}}^{i{\theta }_{0}}{h}^{\prime }\left(z\right)\ge \mathrm{cos}{\theta }_{0}\frac{1-{|z|}^{k-1}}{1+{|z|}^{k-1}}+|{g}^{\prime }\left(z\right)|,$

$|h\left(z\right)|+|g\left(z\right)|\le {\int }_{0}^{|z|}\left(\mathrm{cos}{\theta }_{0}\frac{1+{t}^{k-1}}{1-{t}^{k-1}}+|\mathrm{sin}{\theta }_{0}|\right)\text{d}t$ (2)

$||h\left(z\right)|-|g\left(z\right)||\ge {\int }_{0}^{|z|}\mathrm{cos}{\theta }_{0}\frac{1-{t}^{k-1}}{1+{t}^{k-1}}\text{d}t.$ (3)

$\mathrm{cos}{\theta }_{0}\left[-|z|+2\mathrm{arctan}|z|\right]\le |f\left(z\right)|\le \mathrm{cos}{\theta }_{0}\left[-|z|+\mathrm{log}\left(1+|z|\right)/\left(1-|z|\right)\right]+|\mathrm{sin}{\theta }_{0}||z|.$

NOTES

*第一作者。

[1] Duren, P. (2004) Harmonic Mappings in the Plane. Cambridge University Press, Cambridge, 20.
https://doi.org/10.1017/CBO9780511546600

[2] Lewy, H. (1936) On the Non-Vanishing of the Jacobian in Certain One-to-One Mappings. Bulletin of the American Mathematical Society, 42, 689-692.
https://doi.org/10.1090/S0002-9904-1936-06397-4

[3] Duren, P. (1982) Univalent Functions. Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 47.

[4] Alexander, J.W. (1915) Functions Which Map the Interior of the Unit Circle upon Simple Regions. Annals of Mathematics, 17, 12-22.
https://doi.org/10.2307/2007212

[5] Noshiro, K. (1934) On the Theory of Schlicht Functions. Journal of Faculty of Science, Hokkaido Imperial University. Series I. Mathematics, 2, 129-155.
https://doi.org/10.14492/hokmj/1531209828

[6] Warschawski, S. (1935) On the Higher Derivatives at the Boundary in Con-formal Mappings. Transactions of the American Mathematical Society, 38, 310-340.
https://doi.org/10.2307/1989685

[7] 乔金静, 黄苗苗, 郭倩南. 一类接近凸解析函数的性质[J]. 河北大学学报自然科学版, 2018, 38(6): 567-571.

Top