Michigan Ma h. J. 58 (2009)
Uni e sal Func ions wi h P esc ibed Ze os
and In e pola ion P ope ies
Luis Be nal-González, An onio Bonilla,
& Ma kus Nieß
Dedica ed o P o esso José Méndez on he occasion o his six ie h bi hday
1. In oduc ion
Roughly speaking, uni e sali y means “exis ence o a dense o bi ”. Thus, in some
sense, uni e sal unc ions a e “uncon olled”. In his pape , we s udy he exis ence
o unc ions ha a e uni e sal wi h espec o di e en ial ope a o s and ha a e, a
he same ime, “con olled” by p esc ibed in e pola ion p ope ies, including p e-
sc ibed ze os and mul iplici ies. P ecise de ini ions a e gi en in wha ollows.
We deno e by N,Z,C, and N0 he se o posi i e in ege s, he se o all in e-
ge s, he complex plane, and he se N∪{0}, espec i ely. I A⊂C hen A◦,¯
A,
and ∂A will s and ( espec i ely) o he in e io , he closu e, and he bounda y o
Ain C.We use C∞ o deno e he ex ended complex plane. Recall ha a domain
is a nonemp y connec ed open subse o C.
Le H() be he linea space o holomo phic unc ions on a domain . In pa -
icula , H(C)is he space o en i e unc ions. Conside he me ic
d( ,h) :=
∞

j=1
1
2j· −hCj
1+ −hCj
( ,h∈H()),
whe e
 −hM:=sup
z∈M
| (z)−h(z)|.
He e {Cj:j≥1}is a ixed exhaus i e sequence o compac subse s o ; ha
is, Cj⊂C◦
j+1(j ≥1)and =∞
j=1Cj.I is possible o selec {Cj:j≥1}so
ha each connec ed componen o C∞ Cjcon ains some connec ed componen
o C∞ ;in pa icula , i is simply connec ed (i.e., i C∞ is connec ed)
hen we can choose e e y Cjwi hou “holes”.
The a o emen ioned me ic dgene a es on H() he opology o uni o m con-
e gence on compac subse s o ;see [5]. In he sequel, we will always conside
he comple e me ic space (H(),d).
Recei ed Ap il 7, 2008. Re ision ecei ed Feb ua y 24, 2009.
The i s au ho has been pa ially suppo ed by he Plan Andaluz de In es igación de la Jun a de An-
dalucía FQM-127 and by MEC G an MTM2006-13997-C02-01. The second au ho has been
pa ially suppo ed by MEC and FEDER MTM2008-05891. Bo h au ho s ha e been pa ially
suppo ed by MEC Acción Especial MTM2006-26627-E.
627
628 L. Be nal-González, A. Bonilla, & M. Nieß
Acco ding o Bai e’s ca ego y heo em, e e y comple e me ic space Xis a Bai e
space; in o he wo ds, he in e sec ion o coun ably many open dense subse s o
Xis also dense in X. In a Bai e space X, a subse is esidual when i con ains a
dense Gδ-subse o X. In pa icula , his applies o (H(),d).
Th oughou his pape , we will use he ollowing no ion o uni e sali y.
De ini ion 1. Le (X,dX)and (Y,dY)be me ic spaces and le L=(Lj)j∈Jbe
a amily o con inuous mappings Lj:X→Y.
(i) An elemen x∈Xis called L-uni e sal i
Y={Ljx:j∈J}.
The se o all such elemen s x∈Xis deno ed by U(L). The amily Lis called
uni e sal i U(L)= ∅.
(ii) Lis called opologically ansi i e i i has he ollowing p ope y: Fo e e y
x∈X,y∈Y, and ε>0, he e is a z∈Xand a j∈Jsuch ha
dX(x,z)<ε and dY(y,Ljz)<ε.
In ac , he las de ini ion can be easily ex ended o he se ing o opological
spaces, bu such a gene ali y will no be needed he e. I X=Yand L:X→X
is a con inuous sel -map, hen Lis said o be uni e sal ( opologically ansi i e,
esp.) i he amily L={Ln:n≥1}o i s i e a es is uni e sal ( opologically
ansi i e, esp.). I X,Ya e opological ec o spaces and he Lj(o L,i wea e
dealing wi h sel -maps) a e linea , hen i is cus oma y o say hype cyclic ins ead
o uni e sal. Reade s in e es ed in hese concep s a e e e ed o he su eys [10]
and [13].
In 1952, MacLane [14] s a ed ha he e exis en i e unc ions ϕsuch ha he
se o de i a i es {ϕ(n) :n∈N}is dense in (H(C),d) o , equi alen ly, ϕ∈U(D)
o D={Dn:n∈N}, whe e Dis he di e en ia ion ope a o on H(C)gi en by
D = .In 1994, He zog [12] posed he ollowing ques ion: Which addi ional
p ope ies o elemen s o Xa e compa ible wi h uni e sali y? Fo U(L) esidual
and A⊂XaGδ-subse , he p o ed ha unde ce ain condi ions on Aand L(see
Sec ion 2) he se A∩U(L)is esidual in A.
By using his heo em, He zog de i ed he exis ence o D-uni e sal unc ions
ha ing a ze o- ee q h and (q +1) h de i a i e (q ∈N0). This esul was ex ended
by he i s au ho (see [1] and [2, Thm. 12]) o in ini e-o de di e en ial ope -
a o s (D) =∞
n=0anDn, whe e Dis again he di e en ia ion ope a o (wi h
D0=I, he iden i y ope a o ) and (z) =∞
n=0anznis an en i e unc ion o
subexponen ial ype; ha is, gi en ε>0, he e is a posi i e cons an A=A(ε)
wi h |(z)|≤Aeε|z| o all z∈C.Recall ha an en i e unc ion is said o be o
exponen ial ype i he e a e posi i e cons an s Aand Bwi h |(z)|≤AeB|z| o
all z∈C.O cou se, e e y en i e unc ion o subexponen ial ype is o exponen-
ial ype. By an ope a o we mean a con inuous linea sel -map on a opological
ec o space. I is no di icul o see ha i is o subexponen ial ype hen (D)
de ines an ope a o on H() (and on H(C), assuming only ha is o exponen-
ial ype).
Uni e sal Func ions wi h P esc ibed Ze os and In e pola ion P ope ies 629
In [15,Thm. 3.3], he hi d au ho showed he exis ence o D-uni e sal unc ions
ha sol e a gi en in e pola ion p oblem in C.Independen ly and wi h a di e -
en app oach, Cos akis and Vlachou [7] a i ed a he same conclusion o any
simply connec ed domain. Mo eo e , in [15, Thm. 2.3] i is p o ed ha he e a e
MacLane-uni e sal en i e unc ions ha ing ze os a p esc ibed poin s wi h p e-
sc ibed o de s. On he o he hand, a celeb a ed esul due o Gode oy and Shapi o
(see Sec ion 2) asse s he uni e sali y on H(C)o e e y di e en ial ope a o (D)
as desc ibed he e ha is no a mul iple o he iden i y. Recen ly, he i s au ho [3]
demons a ed he exis ence o (D)-uni e sal holomo phic unc ions wi h gi en
in e pola ion p ope ies.
Ou aim in his pape is o p o e he exis ence o holomo phic unc ions on a
simply connec ed domain  ha simul aneously sa is y he ollowing condi ions:
(a) is (D)-uni e sal;
(b) has ze os (only) a he poin s o a gi en subse o , wi h p eassigned o de s;
and
(c) assumes p esc ibed alues a p esc ibed poin s.
This will be accomplished in Sec ion 3.
The combina ion o uni e sal Taylo se ies wi h he p ope y (b) o (c) has been
conside ed by Cos akis [6]. His imp o emen on He zog’s heo em is also one o
ou auxilia y esul s (see Theo em 3).
2. P elimina y Resul s
This sec ion is de o ed o es ablishing a numbe o s a emen s ha will be needed
in he p oo o ou main esul . We begin by p esen ing he ollowing e sion, due
o G osse-E dmann [11], o he well-known Bi kho ansi i i y heo em.
Theo em 2. Le Abe a nonemp y Gδ-subse o a comple e me ic space, le Y
be a sepa able me ic space, and le L=(Lj)j∈Jbe a amily o con inuous map-
pings Lj:A→Y. Then he ollowing asse ions a e equi alen :
(i) he e is a dense se o elemen s o A ha a e L-uni e sal;
(ii) Lis opologically ansi i e.
I ei he condi ion holds hen he se U(L)is a dense Gδ-se , and so is esidual,
in A.
Recall ha a Polish space is a sepa able comple e me ic space. Nex , we s a e
he He zog c i e ion [12] conce ning inhe i ed uni e sali y—mo e p ecisely, he
(sligh ly imp o ed) e sion due o Cos akis [6].
Theo em 3. Assume ha Xis a Polish space and ha Yis a sepa able me ic
space. Also le dX,dYbe he co esponding me ics. Le Ln:X→Ybe a se-
quence o con inuous unc ions wi h U({Ln}) esidual in X. Fo any B⊂X,le
Ln|Bbe he es ic ion o Ln o B. Conside a sequence {Bk}k∈No Bai e spaces
ha a e subse s o Xsa is ying
630 L. Be nal-González, A. Bonilla, & M. Nieß
A:=
k∈N
Bk= ∅, (1)
Bk∩U({Ln})is esidual in Bk.(2)
I , in addi ion,
lim
k→∞ sup
n∈N
in
h∈A(dX(bk,h) +dY(Lnbk,Lnh)) =0 (3)
holds o e e y sequence {bk}k∈Nwi h bk∈Bk, hen he se U({Ln|A})is esidual
in A.
We also p esen he ollowing esul abou he “in e nal con ol” p ope y o di -
e en ial ope a o s (see [4]). We omi i s easy p oo , which is based on he Cauchy
in eg al o mula o de i a i es.
Theo em 4. Le ⊂Cbe a domain and le be an en i e unc ion o sub-
exponen ial ype. Assume ha K,La e compac se s in Cwi h L⊂K◦.Then
he e exis s a cons an C=C(K,L) ∈(0, +∞)such ha
(D) L≤C K o all ∈H().
The p e iously men ioned heo em o Gode oy and Shapi o s a es ha i is an
en i e unc ion o exponen ial ype hen he di e en ial ope a o (D) is uni e -
sal on H(C)[8, Sec. 5]. By es ic ing he class o ope a o s, we may ex end he
esul o all domains wi hou holes. Speci ically, we ha e he ollowing asse ion,
which can be ound in [2, Thm. 8].
Theo em 5. Le be a noncons an en i e unc ion o subexponen ial ype and
conside he ope a o S=(D):H() →H(),whe e is a simply con-
nec ed domain o C.Then Sis uni e sal. In ac , U(S)is esidual in (H(),d)
o S={Sn:n∈N}.
The inal lemma in his sec ion combines in e pola ion and app oxima ion. I is
a kind o He mi e in e pola ion using exponen ial unc ions ins ead o polyno-
mials. The esul imp o es [3, Lemma 2.1]. By ea(a ∈C)we deno e he unc ion
ea(z) :=exp(az), and span X0will s and o he linea span o a subse X0o a
ec o space.
Lemma 6. Assume ha L,Ka e compac subse s o a domain ⊂Cwi h
L⊂K◦, ha a1,...,ana e di e en poin s in L, ha mis a na u al numbe , and
ha Gis a nonemp y open subse o C.Then he e exis a posi i e cons an M=
M(L,K,a1,...,an,G,m) and a ini e se o unc ions
{αj,k:j=1, ...,n;k=0, ...,m−1}⊂span{ea:a∈G},
depending only on G,m,and he poin s a1,...,an, ha sa is y he ollowing p op-
e y. Fo each pai o unc ions ,h∈H(), he unc ion ϕde ined by
ϕ(z) =h(z) +
n

j=1
m−1

k=0
( (k)(aj)−h(k)(aj))αj,k(z)
Uni e sal Func ions wi h P esc ibed Ze os and In e pola ion P ope ies 631
sa is ies:
(a) ϕ∈H();
(b) ϕ(σ)(aj)= (σ)(aj)(j =1, ...,n;σ=0, ...,m−1);
(c) ϕ− L≤Mh− K.
P oo . We can assume ha G= C, so we choose a poin c∈Gand selec a posi-
i e numbe dsa is ying
d< 1
m(n +1)in {|z−c|:z∈C G}(4)
and
d< min
j,l∈{1,...,n}
j=l
1
|aj−al|.(5)
We de ine
,j(z) :=
n

l=1
l=j
(ed(z −al)−1)m(j =1, ...,n),
βj,k(z) :=ec(z −aj)(ed(z −aj)−1)k
k!dk
,j(z)
,j(aj)
(j =1, ...,n;k=0, ...,m−1).
F om (5), i ollows ha
0<d|aj−al|<1<2π
o all j,l∈{1, ...,n}wi h j= l,so,j(aj)= 0 o all j∈{1, ...,n}.Also, an
easy calcula ion shows ha
β(σ)
j,k(a )=1i =jand σ=k,
0i = jo σ<k,
whe e σ∈{0,1, ...,m−1}is always assumed. We ha e no in o ma ion abou he
alues o β(σ)
j,k(a ) o =jand σ>k.Hence, o each j∈{1, ...,n}we se
αj,m−1(z) :=βj,m−1(z) and
αj,k(z) :=βj,k(z) −
m−1

ν=k+1
β(ν)
j,k(aj)αj,ν(z) (k =0,1, ...,m−2),
whe e he las exp ession makes sense only i m≥2.By induc ion we ob ain
α(σ)
j,k(a )=1i =jand σ=k,
0i = jo σ= k.
Obse e ha each unc ion βj,k, and so each unc ion αj,k, is a ini e linea com-
bina ion o unc ions o he o m ec+sd wi h 0 ≤s < m(n +1). Bu each poin
c+sd is in Gbecause o (4). Hence, he unc ions αj,ka e in span{ea:a∈G}.
Le L,Kbe compac subse s as in he s a emen . By using Theo em 4 (wi h
(z) =zk)o simply he Cauchy es ima es, we ob ain

632 L. Be nal-González, A. Bonilla, & M. Nieß
g(k)L≤MkgK(k ∈N0,g∈H())
o some posi i e cons an Mk ha is independen o g. Now we se
M:=1+
n

j=1
m−1

k=0
Mkαj,kL.
Finally, i we ix holomo phic unc ions ,hon and de ine he co esponding
unc ion ϕas in he s a emen , hen p ope ies (a), (b), and (c) a e ob ious.
3. Main Resul
Suppose ha is a domain in C, and le w={wk}k∈N,γ={γk}k∈N,m=
{mk}k∈N, and β={βk}k∈Nbe sequences sa is ying: w⊂,γ⊂,β⊂C {0},
m⊂N,w∩γ=∅; he poin s wk,k∈N(as well as he poin s γk,k∈N)a e pai -
wise dis inc ; and nei he wno γha e accumula ion poin s in . To each such
se o sequences we can associa e he se A=A(w,m;γ,β) de ined by
A:={ ∈H() : (wk)=0 o o de mk, (γk)=βk(k ∈N),
and (z) = 0i z∈ w}.(6)
In o he wo ds, Ais he se o holomo phic unc ions in wi h p esc ibed ze os
and in e pola ion condi ions (co esponding o w,m,γ,β). Fi s , we no e in he
ollowing p oposi ion ha Apossesses good opological p ope ies.
P oposi ion 7. The se Ade ined in (6) is a nonemp y Gδ-subse o H();in
addi ion, i is a Bai e space when endowed wi h he compac -open opology in-
he i ed om H().
P oo . By he Weie s aß ac o iza ion heo em (see e.g. [18,Thm.15.9]), we know
ha he e exis s a unc ion h∈H() such ha
h(wk)=0 o o de mk(k ∈N)and h(z) = 0i z= wk(k ∈N).
Each γkis di e en om all he wk,soβk/h(γk)is well-de ined. Fo each k∈N,
le αkbe a ixed loga i hm o βk/h(γk). By [18,Thm. 15.13] he e exis s a unc ion
g∈H() wi h g(γk)=αk.Then he unc ion
(z) :=eg(z) ·h(z)
is an elemen o A, hence A= ∅.
Now conside he exhaus i e sequence {Cn:n≥1}gi en in Sec ion 1. Se ing
Mn:={z∈Cn:|z−wk|≥1/n o all k∈N},
we ob ain ha Ais he in e sec ion o he open se s
An:={ ∈H() :| (ν)(wk)|<1
n o 0 ≤ν<m
k, (mk)(wk)= 0,
| (γk)−βk|<1
n o 1 ≤k≤n, minz∈Mn| (z)|>0},
Uni e sal Func ions wi h P esc ibed Ze os and In e pola ion P ope ies 633
whe e n≥1.So, Ais a Gδ-subse o (H(),d). Finally, since Ais a Gδ-subse
in a comple e me ic space, Alexand o ’s heo em (see [17]) gua an ees ha he
opological space Ais comple ely me izable. Hence, i is a Bai e space.
Now, we suppose ha is a simply connec ed domain. Be o e es ablishing he
p omised esul on in e pola ion in i s ull s eng h, we p esen he ollowing “dis-
c e e” e sion o i , which will be used in he p oo o Theo em 9.
Lemma 8. Le Lbe a compac subse o ,le w1,...,wn1and γ1,...,γn2be
pai wise dis inc poin s in L◦,and le m1,...,mn1∈Nand β1,...,βn2∈C {0},
whe e n1,n2∈N.We de ine
B:={ ∈H() : (wk)=0o o de mki k=1, ...,n1,
(γk)=βki k=1, ...,n2,
(z) = 0i z∈L {wk:k=1, ...,n1}}.
Endow Bwi h he compac -open opology inhe i ed om H(). Le be a non-
cons an en i e unc ion o subexponen ial ype, and le S=(D) and S=
{Sn:n∈N}.Then U(S)∩Bis a dense Gδ-subse o B.
P oo . In a simila manne as o he se Ain P oposi ion 7, we deduce ha Bis
also a Gδ-subse o H(). Acco ding o Theo em 2, i su ices o show ha Sis
opologically ansi i e. We he e o e ix ∈B,g∈H(), a compac se K⊂,
and a numbe ε>0.We ha e o show he exis ence o some ϕ∈Band some
N∈Nwi h
sup
z∈K
|ϕ(z) − (z)|<ε (7)
and
sup
z∈K
|SNϕ(z) −g(z)|<ε. (8)
Since ≡ 0 and is simply connec ed, we can ind a compac se L1⊂
sa is ying he ollowing p ope ies:
•C L1is connec ed;
•K∪L⊂L◦
1;
•∂L1is a egula Jo dan cu e; and
• is ze o- ee on ∂L1.
Thus
d:=min
z∈∂L1
| (z)|>0.
I has u he ze os on L◦
1(apa om w1,...,wn1), we deno e hem by ζ1,...,ζn3.
Deno e by 1,..., n3 hei espec i e o de s.
By hypo hesis, he en i e unc ion is noncons an , so he open se
G:={z∈C:|(z)|<1}
is nonemp y. Choose a compac subse L2⊂wi h L◦
2⊃L1, and le
634 L. Be nal-González, A. Bonilla, & M. Nieß
M=M(L1,L2,w1,...,wn1,γ1,...,γn2,ζ1,...,ζn3,G,
max{m1,...,mn1, 1,..., n3})
be he posi i e cons an p o ided by Lemma 6. Acco ding o Theo em 5, he e
exis s an S-uni e sal unc ion h∈H() wi h
h− L2<1
Mmin{d,ε}.
By Lemma 6 one can ind a unc ion ϕ=h+α∈H(), wi h α∈span{ea:
a∈G}, ha sa is ies
ϕ(σ)(wk)= (σ)(wk)=0(σ =0, ...,mk−1;k=1, ...,n1),
ϕ(σ)(ζk)= (σ)(ζk)=0(σ =0, ..., k−1;k=1, ...,n3),
ϕ(γk)= (γk)=βk(k =1, ...,n2),
and
ϕ− L1≤Mh− L2<min{d,ε}.(9)
Condi ion (7) ollows di ec ly om (9) and om he ac K⊂L1.By (9) we ha e
|ϕ(z) − (z)|<d≤| (z)| o all z∈∂L1,
so we conclude om Rouché’s heo em [18, Chap. 10] ha he unc ions and ϕ
ha e he same numbe o ze os, coun ing mul iplici ies, in L◦
1.Owing o he in e -
pola ion p ope ies o ϕ(obse e ha he o de o he ze o wk,ζk o ϕis a leas
mk, k, espec i ely), i ollows ha ϕ∈B.
I emains o show he exis ence o an N∈Nsuch ha (8) is ul illed. By de ini-
ion, |(a)|<1 o e e y a∈G. Since (D)ec=(c)ec o all c∈C,weha e
(D)nec=(c)nec o e e y n∈N.Hence (D)nea→0(n →∞)compac ly
on Cwhene e a∈Gand, as a esul ,
(D)nα→0(n →∞)compac ly on C o e e y α∈span{ea:a∈G}.
In pa icula , he e is an N1∈Nsuch ha
SnαK<ε/2(n ≥N1).
Because his S-uni e sal, he e is an N≥N1wi h SNh−gK<ε/2.Finally,
since ϕ=h+αwi h α∈span{ea:a∈G}, he linea i y o S oge he wi h he
iangle inequali y d i es us o (8), as equi ed.
We a e now eady o s a e ou main esul .
Theo em 9. Assume ha ⊂Cis a simply connec ed domain and ha is
a noncons an en i e unc ion o subexponen ial ype. Le S=(D) and S=
{Sn:n∈N}.Suppose ha w={wk},γ={γk},m={mk},and β={βk}a e se-
quences as in he beginning o his sec ion, and suppose he se A=A(w,m;γ,β)
is de ined as in (6). Then he se A∩U(S)is esidual in A. In pa icula , in A
he e is a dense Gδ-subse all o whose unc ions a e (D)-uni e sal.
Uni e sal Func ions wi h P esc ibed Ze os and In e pola ion P ope ies 635
P oo . The essen ial ool o he p oo o his heo em will be Theo em 3. The e-
o e, le X:=(H(),d) =:Y. Recall ha (H(),d) is a Polish space, so i is a
sepa able me ic space as well. Le Ln=Sn(n ∈N). F om Theo em 5 we know
ha U(S)is esidual in (H(),d).
Wi hou loss o gene ali y, we can assume ha he exhaus i e sequence {Ck:
k∈N}o compac se s de ining he me ic do H() (see he In oduc ion) sa is-
ies ha he e a e wo s ic ly inc easing sequences {j1(k)}∞
1,{j2(k)}∞
1⊂Nsuch
ha Ck∩w={w1,...,wj1(k)}=C◦
k∩wand Ck∩γ={γ1,...,γj2(k)}=C◦
k∩γ.
We de ine
Bk:={ ∈H() : (w
j)=0 o o de mji j=1, ...,j1(k),
(γ
j)=βji j=1, ...,j2(k),
(z) = 0i z∈Ck {wj:j=1, ...,j1(k)}}.
Ob iously, he in e sec ion o he se s Bkis exac ly he se A, which is nonemp y
by P oposi ion 7. An a gumen simila o he one gi en in he p oo o ha p opo-
si ion shows ha each Bkis a Bai e space. Now, Lemma 8 shows he co ec ness
o condi ion (2) in Theo em 3.
In o de o apply Theo em 3, i emains o p o e ha o e e y sequence {bk}k∈N
wi h bk∈Bkwe ha e
lim
k→∞ sup
n∈N
in
ϕ∈A(d(bk,ϕ) +d(Snbk,Snϕ)) =0.(10)
Le k∈Nand ∈Bkbe ixed. Hence, he ze os o in Cka e exac ly gi en by
he poin s wj(o o de s mj)wi h j∈{1, ...,j1(k)}.Choose a Jo dan subdomain
Uk⊂such ha Uk⊃Ckand
w∩(Uk Ck)=∅=γ∩(Uk Ck),
which is possible because nei he wno γha e accumula ion poin s in .
Le g∈A. Since /g is holomo phic and ze o- ee in Uk, he e exis s a unc ion
φholomo phic in Uksuch ha
(z)
g(z) =eφ(z) (z∈Uk). (11)
Fo z=γjwi h j≤j2(k), he le -hand side o (11) is 1 and so φ(γ
j)=µj·2πi
o some µj∈Zi j≤j2(k). Nex , le h∈H() wi h ze os exac ly a he poin s
γjwi h j>j
2(k) and wi h h(γj)=1i j≤j2(k). Tha such a unc ion exis s
can be p o ed by an a gumen simila o he one used in he p oo o P oposi ion 7.
Fu he mo e, we se
c1:=h¯
Vk,c2:=g¯
Vk,c3:=exp(φ¯
Vk+1),
whe e Vkis again a ixed Jo dan domain, his ime sa is ying Ck⊂Vk⊂¯
Vk⊂Uk.
Suppose ε>0.By Walsh’s heo em on simul aneous app oxima ion and in e -
pola ion [19], he e exis s a polynomial psa is ying




φ
h−p


¯
Vk
<minε
c1·c2·c3
,1, 1
c1