ANNALES UNIVERSITATIS
MARIAE
CURIE SKŁODOWSKALUBLIN
POLONIAVOL. Xl.ltl, 1__________ ______________SECT1O A___________ ______________________ 1989 Zakład Zastosowań Matematyki
Instytut Teorii Rozwoju Społeczno Ekonomicznego UMCS
F. BOGOWSKI ,
CZ.BURNIAK
On
the Relationship
between the Majorization ofFunctions and the
Majorization ofDerivatives in Certain Classes
of Holomorphic Functions
O zależności między
majoryzacją
funkcji a majoryzacjąpochodnych
w pewnych klasach funkcji holomorficznychAbstract. In this paper we investigate the relationship between the majorization of functions and the majorization of derivatives in the class H of functions in the unit disc K, satisfying the condition
Re|(l_
z2)M} >0,
as well as in the class H* of close-to-star functions.
The results obtained are sharp.
l.Introductlon.
Let C
denote the complexplane,
Kr ={z G C
: |z|< R},
Ki = K and f , F betwo
holomorphic functionsin
thedisc
Krsuch that
/(0) = F(0). We saythat
fis
majorized by Fin
thedisc
Kr and write f C Fin
Krif |/(z)| <
|F(z)|for every
z£
Kr. It means that
thereexists
in Kra
holomorphic function$
suchthat
|$(z)| <1
for z G Kr and(1) /(z) = *(z)-F(z)
for
everyz £ K
r.
Denote
by Ti, Ftwo fixed andcompact
classesof holomorphic functions
in the discK.
Supposethat f
GTi,
F£
Fandf <
F inK.
Letr0
G (0; 1) denote the greatestnumber
for which thefollowing implication
/<F
in
F=>/'<F'in
Rroholds for
every pair offunctionsf G TC,
F G F. The number ro = r0
(R,.F)*
sca^
e<f theradius
ofmajorization
of derivatives for the class F. Thedetermination of ro
invarious subclasses of univalent
functionsin
the discK
wasinvestigated by
manyauthors.
Among
othersZ.
Lewandowski [4] provedthat
r0
(H,S)
=2 —
a/3,where S
denote theclass
offunctions Fholomorphic
and univalentin
the discK
such thatF(0)
=0,
F'(0) = 1.This
problem can begeneralized
in thefollowingway. We willdetermine
thesmallest possible
number T(r)=
T(r, If,F) such that for every
pairof
functions f,F(f €
W,F € F)
theimplication
/<F in
K=*
|/'(z)|< T(r) •
|F*(z)|for |z( =
r< 1
holds.T(r) can be
defined as follows
(2)
Let us note that
theradiusof majorization
ofderivatives r0 = sup {T(r) <
1} .re(O)l)
It follows from the définition
of
the radius ofmajorization of
dérivatives andT(r)
that it must be |z| <r/-,
wherer? is
theradius of
localunivalencein
theclassF i.e.
r? —
sup{|z| <
rG
(0;1)
: F'(z) / 0for
ailF G
F] .Among others
the following result wasobtained(J.- Janowski ,
J. Stankiewicz[3])
1 for
rG (0;
2— \/3)
T(r,W,S)= { 4r
2 +
(1- r)<
for
rG (2- \^;1) .
4r(l—r)2
2. The
majorization of derivativesin class H. Denote
by H theclass of
functions F,holomorphic in
the discK, satisfying
thecondition
(3)
Re
{(l-z’
)^}>0,
zGF
and
such thatF(0)
=0,
F'(0) = 1.If
the coefficients of the functionF
satisfyingcondition (3) are real,
then the functionis
typicallyreal. The function F
holomorphic inK
and suchthatF(0)
=0, F
’(0)
= 1,is
saidto betypically
realif
it takes the realValueson thesegment
(—1;
1) ofthe realaxis and satisfies theconditionIm z •
ImF(z) >
0for z G
K \(—1;
1). The classof
typically realfunctionswill
be denotedbyTR.
The
class H containsthentheclass TRoitypically real functions.The condition (3)
canbe writteninthe equivalentform F(*)=
j“» H*)’
(4)
On the Relationship between the Majorization of Functions... 3 where p € P
,
P being the classof functions p
holomorphic in K andsuch
thatRe
p(z)>
0for z
GK
andp(0)=
1.Theorem
1 [1]. The. radius oflocal univalence in the class His
equalRemark . The
radius of
local univalence ruis simultaneously
the radius of univalence (i.e.in
discK
r„ every function FGH is
univalent)as well as
theradius
ofstarlikeness in
theclass
H (i.e.in
disc Kr
„ everyfunction F G H satisfies
the conditionRe >
0 ).F(*J
Theorem
2.
IfFG H thenfor z G
K,rHr(l-r-*)
(5)
F(z)
F'(z)l - (1 — r2) — 2r(l + r2) ’
The estimation
is sharp
and theequality
holdsfor the
functionF
of the formz 1
+iz(6) F(z) =
1 —
z2 1
—izr = ?
at the point z =
ir.
Proof.
It follows from
thecondition
(4) thatzF'(z) 1 + z2 _zp\z)
F(z) 1
- z2 p(z)where
pG
P.Making use
ofthewell-known sharp
estimationI p(z)
~ 1 -
l*l2
for pGP
we obtain(7)
lzF'(z) I F(z) l+z2l 2|z|l-z2l
“ l-|z|2
FYom(7)
theinequality (5) follows.
A simple calculation shows
that
the function oftheform(6) for z
=irgivestheequality in
(5).Theorem 3.
Let f G
H, FG
H.
Iff
<£. F in the disc K and|z| =
r < rH = HH/5 l + \/5 2then
(8) |/'(z)| < T(r) • |F'(z)| ,
where
(9) T(r)
’ 1
< [(1 -r2)2 -2r(l +r2)]2 +4r2(l +r2)2 4r(l + r2)[(l — r2)2 — 2r(l + r2)]
for
re
(0,r0) for r
€ (r0,rH) .The
number ro = 1
+\/2
— \/2+2\/2
«0,217 is the
uniquepositive
rootof
the equation(10) (1 — r2)2 — 2r(l + r2)
2r(l + r2) in the
interval
(0,r//).The result is
sharp.For
r 6 (0, r0
) andfor
everypairof functions f, F such that f(z) =
F(z),F&H, wehave T(r) =
1.For
rj
€ (ro,r/y) the equalityin
(8) holds at the point z=
irj for the pairof
functions f, Fsuch that/(z)
=$(z)-F(z), where F
isgiven
by(6)
and(11)
z —
irj$
(?) =1
+A/(ri)
+M(n)
z
—?ri 1 +
irjz(12) M(r) (1 - r2)2 - 2r(l + r2)
2r(l + r2)
Proof,
If f
<C F in thedisc K, then
there exists afunction |$(z)| < 1 for
zG
K suchthat
/(z)
= 4?(z)•
F(z) for z £A'
. Hence(13)
F'(z)4>'(z) F(z)
F'(z)
It
is
known(cf. see [2]p.319) that
i
- l*(*)l2
l-|z|
2 for
z £K .
(14)
!*'(*)! <On the Relationship between the Majorization of Functions... 5 Taking account of the
inequalities
(5) and (14) in (13)we obtain
(15) |/
,(*)|
<-l*(*)l2
lf'(z)|-
2M(r) +!$(*)! +
l2M(r)
’ whereAf(r) is
given by theformula
(12).Let
z be fixed, |z| = r,then
theright-hand
sideof
inequality (15)is
a function ofthevariableu
= |$(z)|,u € (0,1),
+ u + 1
(16)
*(«) =
2M(r) 2A/(r) '
It should
benoted that Af(r) is
a decreasing function in theinterval
(0,1). If r €(O,r
o),
where r0 is
the rootof
theequation (10),
then M(r) > 1.Then 4'
is increasingin
the interval (0,1) and at thepoint u
= 1it attains
the greatestvalue
4>(1) = 1.
If
r G (ro.r//), then0
=M(
th)<
Af(r)< 1
and at
thepoint
uq = A/(r) the function "kattains
thegreatestvalue
equal(17) = (1 — r2)2 — 2r(l + r2)2 + 4r2(l + r2)2 1 °> 4r(l + r2)[(l - r2)2 - 2r(l + r2)]
It
is easily seenthat
for|z| = r G
(0, ro) andfor
every pair offunctionsf,F,
F €H
suchthat
/(z)=F(z)
An elementary
calculation
showsthat
for rj 6(ro,rn)
and for the pairof
functions f,F, where F is
given by formula(6),/(z) = 4>(z)-F(z)
and <f> defined by(11) for
z = iri we havewhere ’
I'(uo) is defined
by (17).3.
The majorization
ofderivatives
inthe class
ofclose-to-star
functions. LetS*
= (G
gH: Re ’
G'(z)G(z) >
0for zG
A', G(0)= 0,
Itis
thewell-known class
ofstarlike
functions.Tin' function
F
hnloinorphic indiscA'
and suchthat F(0)
=0, F'(0)
= 1,is said to
be dose to star ifthere exists
afunction
G6
S*such
that( IS) Re
>
0 for2
6A' .
G(z)
The classofclose
to
starfunctionsis
denotedby
H*.It is
easyto observe
thatif
G(z)= --- ,
G e5*,
then thecondition
(18)takes
theform
Re
1-22
,<(£)■
z
It
meansthat H C H*.
Theorem 4.
The radius ofstarlikeness inthe class
H*zF'(z)
*
=
supj\z
|<
r€ (0,1)
:Re ? >
0for
eachfunction F
€H'
} =2— \/3
.r t F( Z) F(z) J
Proof. From the condition(18) it
follows
that F(2)G(z) =
p(
z) , peF.
Hence
zF'(z) _
zG'(z)
zp'(z)F(z) G(z)
+i>(z)
Usingthewell-known
andsharp estimations
> Lu
G(z) - 1 +
r’ =
Ges- andweget
(19)
, r = W. pep p(z)
1 -
r2„
zF'(z)(1 - r)
2-
2r Re“
fvT - i-p " •
Thisestimationis sharp.
Theequalityholds
for the function FZ
1
—2
(20)
F(2) =
(1+z)2 1 + 2
at
thepoint z = r.
Inorder
that
thefunction Fis
starlikeitmust be
1
— r2
(1 — r)2 -2r> 0 .
On the Relationship between the Majorization of Functions...
7
(21)
Hence, it follows that
r*
=2 — \/3 « 0, 268.
Lemma . If
F€ H*
, thenfor |z|
< r*=
2— \/3
we have the »harp estimateF(x)|„
r(l—r
2)
< --- ---
»»• = PI •
IF'(z)
I ~(1
-r)2
—2rThe
equality takes place for
thefunction
(20) at thepoint
z =r.
The
above
estimatefollows
frominequality IF(z)
I-
KC F(z)^(*)and from the estimate (19).
Theorem 5. Let f 6 H,
F €H*
.If f <
Fin the
disc K and.|z| = r <
r*=
2—
\/3, then (22)where
l/'tol < T(r) • |F'(z)|
,T(r) =
1 /orrG(0,r0)
[(
l_r)2_2r]
2+4r2
4r[(i'-r)’-2r] /<”* r e (r0,2 - v^) The
number
ro = 3— \/8
~0,172M
theunique
positiveroot
of the equation(1
-r)2 _2r
(23) L
—£---
=1
in
theinterval (0,2 —
t/3).The
result is
sharp.For
r 6 (0,
ro) the equality in (22)is attained
for every pairof
functionsf, F, F
G H*such that
f(z)= F(z).
If rj G (ro,2 —
\/3) then theequality in (22)is
attainedat.
the point z —»q for
the pairof functions /, F
andsuch that /(z) =
$(z)•
F(z), where(24)
F(z) = 1-z
(1+z)2 1 +
z’
2 e K(25)
$(z) =
2 - fl
1 - rtz + Wi)
1
+Af(ri)
2- rt 1 —
ri2z
G A(1 — r)2 — 2r
(26)
M(r)
2rProof.
From
theequality /(z) =
$(z)•
F(z),
z6 K,
andfrom
theestimations
(14),(21)
it followsthat
(27)
I/'(*)I
„ -w2I
F'(z) I “ 2M(r)
+U+2Af(r)
1 = *(«) ,where
u = |$(z)|,
tt€ (0,1),
zis
fixed,|z|
=r. Proceeding
analogouslyas in
theproof
ofTheorem3 weobtain1 forr
G
(0,3— \/8)
max i’
(u) = <
u6<o;i> [(1 — r)2 — 2r]2 + 4r2 /-
4r[(l-r/-2r|
<<>, r e (3 - VS,2 - VS)
Henceourtheorem
follows.
It
is
easilyseen that for r
6(0,3
—\/8)
theequalityin
(22) is attainedfor
each pairoffunctions
/,F such that /(z)
= F(z), F€ H*.
An
elementary calculation
showsthat
for n £(3- \/8,
2- \/3)
andfor the pairof
functions /,F, f(z}
= $(z)•
F(z),where F
isgiven
byformula (24), $
defined by(25)
weobtain
equalityin
(22) at thepoint
z =rq
.REFERENCES
[1] Bogowski.F. , Burniak, Cz. , On the domain of local univalence and starlikeness in a certain class of holomorphic functions, Demonstr. Math. vol. XX, No 3-4, 1987, 519-536.
[2] Goluzin , G. M. , Geometric Theory of Functions of a Complex Variable, Izd. Nauka, Moscow 1966, (Russian).
[3] Janowski , J. , Stankiewicz , J. , A relative growth of modulus of derivatives for majorized functions, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 32, 4 (1978), 51-61.
[4] Lewandowski , Z. , Some results concerning univalent majorants, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 38, 3 (1964), 13-18.
STRESZCZENIE
W pracy badana jest zależność między majoryzacją funkcji a majoryzacją pochodnych w klasie H, funkcji spełniających w kole jednostkowym K warunek
Re|(l-z
2
)^}>0,
oraz w klasie H* funkcji prawie gwiaździstych.
Otrzymane rezultaty są dokładne.