我們定義一個局部單葉的亞純函數 f 其 Schwarz 導數在 f'(z)≠0
的點為S_f=(f'')(f')'-1/2(f''/f')^2, 而在單極處則訂為 S_f=S_(1/
f). 我們早就知道S_f=0 若且唯若 f 是一個 Mobius 變換. 直接從定義
可得知 S_f 是一個可析函數.相反地, 如果 φ 是一個可析函數, 則存在
一個亞純函數 f 使得 S_f=φ. 早在1949年, Z. Nehari 就證明了 ∣
S_f(z)∣≦ 2/(1-∣z∣^2)^2) 及∣S_f(z)∣≦(π^2)/2 可導出 f 在單
位圓盤上的單葉性.F. W. Gehring 及 C.Pommerenke 更針對第一種條件
作深入的研究, 得到可同胚及可擬保角延拓的充分條件.而我們將注意力
集中在滿足Nehari第二種條件的函數和這些函數的延拓性.而這篇論文主
要是在單位圓盤D中就∣S_f(z)∣≦(π^2)/2 的條件作一檢視. 整個論文
的架構平行於Gehring 及 Pommerenke 的架構. 其中部份的技巧則模仿自
M. Chuaqui 及 B. Osgood 的研究工作.首先, 我們討論了 f 的解析性,
並且對 ∣f'∣及 ∣f∣ 的範圍作一估計.其次, 我們討論了 f 的單葉
性. 接著證明了 f 可連續延拓到 D 的邊界上. 再就此延拓是否一對一分
別討論. 如果是一對一, 則 f 可以擬保角延拓到整個複數平面; 若不是
一對一, f 則與 2/π tan πz/2 Mobius 共軛. 最後, 我們知道如果 f
不是與 2/ π tan πz/2 Mobius 共軛, 則 f 在 D 上滿足 Lipcshitz
條件.

We define the Schwarzian derivative of a function f which is
meromorphic andlocally univalent as S_f=(f''/f')'-(1/2)(f''/f')
^2 at the points where f'(0) is not equal to 0. And we define
S_f(z)=S_(1/f)(z) at the points which are simple poles. It's
shown in early age that S_f = 0 if and only if f is a Mobius
transformation. Directly from the definition, we know that S_f
is analytic. Conversely, suppose that φ is analytic
function, there exists a meromorphic function f such that S_f=
φ. Early in 1949, Z.
Nehari showed that conditions∣S_f(z)∣≦2/(1-∣z∣^2 )^2 and ∣
S_f(z)∣≦(π^2)/2 both imply that f is univalent in the unit
disk D. F. W. Gehring and C. Pommerenke focused their research
on the first condition that Nehari's deduced, and they had the
sufficient conditions that thefunction f can be extended to a
homeomorphic function and a quasiconformalfunction on the plane.

This paper mainly makes an investgation on which properties f
possessesunder the condition ∣S_f∣≦(π^2)/2 in the unit disc
D. The structure of this paper is parallel with the one of
Gehring and Pommerenke. And there are some skills imitated from
the research of M. Chuaqui and B. Osgood. First, we discuss the
analytic property of f under∣S_f∣≦(π^2)/2. We also estimate
the bound of ∣f'∣and ∣f∣. Next, we discuss the univalence of
fand prove that f has a continuous extension to the boundary of
D. Whether this extension is univalent or not give us two
directions. If the extension isone-to-one, f possess a
quasiconformal extension to the whole complex plane.If not, f is
a Mobius conjugate to 2/π tan πz/2. Finally, we prove that if
f is not a Mobius conjugate to f, then f satisfies Lipschitz
condition,which is the special case of Holder's continuity.
We define the Schwarzian derivative of a function f which is