(Tt luận án) Tính chuẩn tắc của họ hàm phân hình một biến và bài toán duy nhất đối với đa thức vi phân

TRƯỜNGĐẠIHỌCSƯPHẠM NGUYỄNVĂNTHÌN TÍNHCHUẨNTẮCCỦAHỌHÀMPHÂN HÌNHMỘTBIẾNVÀBÀITOÁNDUY NHẤTĐỐIVỚIĐATHỨCVIPHÂN TÓMTẮTLUẬNÁNTIẾNSĨTOÁNHỌC THÁINGUYÊN-2016... Sauy chngt icpchitithnvb icnhnysinht

TRƯỜNGĐẠIHỌCSƯPHẠM

NGUYỄNVĂNTHÌN

TÍNHCHUẨNTẮCCỦAHỌHÀMPHÂN HÌNHMỘTBIẾNVÀBÀITOÁNDUY

NHẤTĐỐIVỚIĐATHỨCVIPHÂN

TÓMTẮTLUẬNÁNTIẾNSĨTOÁNHỌC

THÁINGUYÊN-2016

Vàohồi8giờ30ngày10t h á n g 7năm 2016

2 M t sn h l k i u L a p p a n c h o h

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t h e o ccn g d ngtrong vi c nghin cu h m (nh x) phnhnh Ngc l i

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t c a LthuytNevanlinna, ch ngt iphithitlpnhngdngn h lc b nthh a i phh pvitnhhungca

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n a n v U s a o chu n t c v c h o chu n t c v m chu n t c v ikh ngi mc a c c h m trong

u chu n t c v vc h n u t n t ist h c 0<r<1,d chu n t c vyi mz n :|z n |< chu n t c vrvz n →z0,d

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ath coh m t ng qu t.V n n yc gi i quyttrong Chn g

t pt tcc cnh xb o gi c c aUv chu n t c v o chu n t c v ch chu n t c v nh chu n t c v n chu n t c v chu n t c v Lehto

1−|ξ|2

||f||N = supz∈U (1−|z|2)f # (z).

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nhnhfv g k hch ngsaochocc

dng n o,n≥11)thf=c1e cz v g=c2e −cz ho chu n t c vc f=tg, trong chu n t c v chu n t c v chu n t c v chu n t c v chu n t c vc chu n t c vc chu n t c vh chu n t c vng chu n t c vs c1, chu n t c vc2,

- saiphnt h u htc sq u a n tmcanhiunht o nhctrnthgiin

hJ.ZhangvR.Korhonen,A.Fletcher,J.K.Langleyv

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Ngo i c c ph n mu , k tlu n,t ili u tham kh o, Lu

nn c chia lm bachng tngn g viba vnnghincu chnh:

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m c aLappan Cth , ch ng t i thi t l p c cn h l k i u

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r)→ ∞khir− → 1−vRa

(z)= ϕ chu n t c v chu n t c v ( | a +ρz/ϕϕ ϕ(|a|) ( | a | ) | ) hitutrn

f, tal u n c χ (f(z),f(w))≤||f||Nϕ sup ξ∈[z,w] ϕ(|ξ|)|z−w|, trong

f # (z)

||f||Nϕ =sup z∈U .

ϕ(|z|)

im t l p c c h mϕ- chu n t c v c h u chu n t c v n chu n t c v t chu n t c v c chu n t c v r chu n t c v n g chu n t c v h chu n t c v n chu n t c v v chu n t c v chu n t c v chu n t c v t r o n g

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ngkh c, ktqutr nng th i mr ng ktqu c a

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to n tq - sai ph n trong c c b i to n: x cn h d u y n h t h m

ph n d ng[P(f(z))f(qz+c)] (k) , p h chu n t c v n chu n t c v b g i t r k i u H a y m a n

c h o a th coh m kt h pq - chu n t c v s a i chu n t c v p h chu n t c v n chu n t c v d chu n t c v n g [P(f(z))f(qz+ρ c)]

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a =

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1 1

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iukinkhngi mcaa thco hm

imcaf n (f n1 )(t1 )···(f n k ) (t k)−a m c bitnht l m G i sr ng

a) n j ≥t j vim i 1≤j≤k, vl i ≥2vimi1≤i≤q;

ngi mc af n (f n1 )(t1 )···(f n k ) (t k)−a m cb it n h t l m Gis rng

a) n j ≥t j vim i 1≤j≤k, vl i ≥2vimi1≤i≤q;

j=1 j=1

Σ

Σ +

Nhnxt1.7.n h l1 8 vn h l1 1 0 vnn g khithay

f n (f n1 )(t1)···(f n k ) (t k)bia thcohmtngqut

Σk

tj

j I

tv

v I

Σ

j=1

Chn g 2

Mtsn h lkiuLappancho

ng c a l p h m chu n t c Ch ngt ich ng minh m t stiuchunkiuLappancholphmny.Ngoira,chngticngthitlpc c ti uchu n chu n ki u Lappan tn g n g c h o h c h u n t c k h i t

(z)=ϕ chu n t c v chu n t c v ( | a +ρz/ϕϕ ϕ(|a|) ( | a | ) | ) →1 (2.2)

a

R

z∈f −1 (E) ϕ(|z|) z∈f −1 (E\{∞})

Khif l hmphnhnh ϕ -chuntc.

nhl2 6 Choϕ : [ 0 ,1)→ (0,∞)l h m t n g t r n v f l h m ph

nhnht r na n vU G i sr ng E= { a1,a2,a3 }⊂Cg mba

Hq u 2 7 Chof l h mphnhnhtrnan v U v E⊂Cl

ctK⊂ D, t nti E = E(K)⊂Cchabn

tk.Gisr ng v i m i t p compactK⊂D,t n t ia∈C\

(f n f (k)) #(z)≤M v i mi f∈ Fv miz∈K∩{f n f (k) =a}.

z|)= 1 , chu n t c v chu n t c v tanhnc ccktqu

Chn g 3

Sd u y nhtcacchmphn

-saiphnchungnhaumthmnh

Trong chng n y, ch ng t i nghi n c ung d ng c a lthuy t

l t r o n g chng ny

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u f−a v g −a c c n g s k h n g i m t n h c b i H

Nm2014,Q.Ling,L R.JievX Zuxingxt bi ton duynhtcho

−i, ,n),a m−i /=0v ii∈{0,1, ,m},

(i2)f v g t h amnphn g trnhi s R(f,g)=0, trong

va0 0,a1, ,a m−1 ,a m /=0l c chngsp h c

Gis[f n P(f)] (k)v[g n P(g)] (k)chungnhauα(z)−IM, khi f(z)≡tg(z)v itl h

ng s tham nt d = chu n t c v1,trongd= (n+m, chu n t c v chu n t c v chu n t c v chu n t c v, chu n t c vn+m−i, chu n t c v chu n t c v chu n t c v chu n t c v, chu n t c vn), chu n t c va m−i /=0v

ii∈{0,1, chu n t c v chu n t c v chu n t c v chu n t c v, chu n t c vm}hocfvgtham n phngtrnhisR (f,g)=0, trong

R(w1,w2)=w n (a m w m +ρa m−1 w m−1+···+a0 )

—w n (a m w m +ρa m−1 w m−1+···+a0).

i

Nhnxt3.1.i u kinmax{χ1,χ2}<0cthb q u a trongmtlp

kh ng,qvclcc h ng s ph c,q/=0vchokl snguy ndng.Choa th cP(z)

=a n z n +ρa n−1 z n−1+· · ·+a1z+a0v icc h sa0,a1, ,a n−1 ,a n =/0vm lskh ngi

ngn≥m(k+1)+5(n≥m(k+1)+3),khi(P(f(z))f(qz+c)) (k) −a(z)cvhnkh

Nh n x t 3.2.Trongn h l 3 1 4 , k h i m= 1 , c h chu n t c v n g chu n t c v t

i chu n t c v n h chu n t c v n chu n t c v l chu n t c v i chu n t c v k chu n t c v t quc aZhaovZ h a n g

a(z)/≡0lhmphnhnh(hmnguy n)nhcaf,g va thc P(z)=

a n z n +a n−1 z n−1+···+a1z+a0v icchsa0,a1, ,a n−1 ,a n 0. G i m

Nh n x t 3.4.Trongn h l 3 1 6 , k h i m= 1 ,c h chu n t c v n g chu n t c v t chu n t c v i

n h chu n t c v n chu n t c v l chu n t c v i chu n t c v k

ucaZhaovZ h a n g chocchmphnhnhsi uvitvibckhng

KtluncaChn g 3

Trong Chng 3, ch ng t i nghi n c ung d ng c a Lthuy t

m nh C c ktqun ylmr ng c c ktquc a Zhao vZhang

Ktlunvn g h

Lu n n nghi n c u nh ng ng d ng c a Lthuy tNevanlinna

trongbitonhvh mchuntc,bitonduynhtcacchmph nhnhvia