Some asymptotic approximation theorems in real and complex analysis

by Liu, Ming-chit

(Uncorrected OCR) .. _~-~--,,--,_--:...._._------_._-------_.-

Abstract of thesis titled 'SOIl-IE .f~Y1'IP.l.'OTIC APPROXIivIATION THEORE1LS

llJ REAL AND COMPLEX ANALYSIS I submitted by LIU Hll{G-CHIT for

t.he degree of DOCTOR OF PIULOSOPHY at -1:;he UNrVERSITY OF HONG

KONG in February 197.3.

Two par-~s, totally six chapters, comprise the thesis.

The re~in result of the first part is majorant functions of

power serief,. Dr. Y.M. Chen, the author's supervisor and the

author obtained:

Let fez) be defined by the power series

11 h;-l

A z + ah+l z

11-:-2 + ah+2 z

+ ??

where

Isl < 1

and h is some integer

~ 0 ?

Let

11' (A) h

be

the family of anal;ytic functions def'i:iled above such that

If(z)1 ~ 1 in Izi < 1 ~ and let

t1'tef;r) =Arh + la11+11 rh+l + lah+21

h+2

I' + ??

be the majorant function of fez) with Izi = r. We obtained

some es1(imates on the upper bound of -(;he function Bo CA) , where

BoCA) is tiefined as m,(f';r) ~ 1 when I' ~ Bo(A) and fCz) E FoCA); and 9rt(fJr) > 1 for r > BoCA) for some

fCz) E FO(A) ?

The main result of the second part is simultaneous appro-

ximation of real numbers. The author provaiL that

(1) ]'Ol' 8ver'Y rea: 6' and every posH;ive integer N" there is

an in'~eger 11 satisfying

lie 211 1?N--1/2+C(N)

l~Il~I\T .. 1:"1. < ... ,

whe:r:; L i[j em absolute constant, 8 (N) == 1/10g log Nand

11"-11

I "I!

l',~3a:;,n the a.istance from

. ~

?o the nearest integer.

each N ~ Hi

the above inequalitier are true for A == 1.??

(2)

Fo~ integer k ~ 2 let

Ie-I

K == 2 ?

Le-"

"

o be any arbitrary

l'0:o:;iti'.rE! number. lilor any E > 0 there exist some positive

C0113tants C(k., E) and C(O, E) such that for any- real

nurubers 0, ?and integer N ~ 1 there exists an in-teger

1'1 sD.tisf"y:L1g

? I 1 '" n :::; N , rl I \ IIOn211 < C( 0, E)N-1/(7+o)+? , lI?2'1 < C(o, E )N-1/(7+o )+E; II I lI0l1'~'1 < C(k, )i'rl/C3K+l)+E 111>11 kll < C(k, E )N-l/(3K:.l)+? Ii I E 1 , -