495
Internat. J. Math. & Math. Sci.
Vol. i0, No. 3 (1987) 495-502
ON SOME CLASSES OF ANALYTIC FUNCTIONS
KHALIOA I. NOOR lind HAILA MADIFER
Mathematics Department, Science College
of Education for Girls, Sitteen Road,
Malaz, Riyadh, Saudi Arabia.
(Received April 18, 1986)
ABSTRACT:
Let
be any numbers and let
ml,m 2
Vml,m 2
functions of analytic in the unit disc E={z:
zl<l}
be the class of
for which
m
I
(s ’(z))
1
f’(z)
m
(s ’(z)) 2
2
Moulis [i]
are analytic in E with S (0)=(S’ (0)=i
and S
2
1
2
1
gave a sufficient condition and a necessary condition on parameters
where S
m
S
I
1
and m
-
and S
2
2
for the class V
ml=
e
to consist of univalent functions if
In fact
are taken to be convex univalent functions in E.
he proved that if feV
k+2
ml,m 2
-is
m
I
(l-o)cose
m
m
where S
2
2
and S
1
k-2
---(l-0)e
are convex and
2
21ml+m21 _< i,
cos,
then f is univalent in E.
In this paper we consider the class V
ml,m 2
in more general way and
show that it contains the class of functions with bounded boundary
rotation and many other classes related with it.
results,
Some coefficient
arclength problem, radius of convexity and other problems
are proved for certain cases.
Our results generalize many previously
known ones.
KEY WORDS AND PHRASES. Univalent functions, boundary rotation, radius of convexity.
1980 MATqEMATICS SUBJECT CLASSICATION CODE, 30C32, 30C34
i.
INTRODUCTION.
Let V
E={z:
k(0)
Izl<l},
IR
be the class of all functions f, analytic in
f’(0)=l, f(0)=0, f’(z)# such that for z=re
e
e
zf’
where k>2, 0<0<I,
(z))’-0coseld@<k
1-0
real and
lel
<
2
cose,
0<r<l
K.I. NOOR AND H. MADIFER
496
e
For
has been introduced and studied by Moulis in [i].
e
and
introduced and studied in [2].
e=O
0=0
k
of functions with bounded boundary rotagive us the well known class V
k
tion first introduced and discussed by Paatero [3] and Lowner [4]. FuncThe class V
k(0)
0=0, we obtain the class V
V
in
tions
V(0) may not possess boundary rotation.
class T(0) of analytic functions which is a
and
Also a
of
V(0)
[5].
has been discussed in
Tk(O)
f’(O)-i is in
if for zEE, there exists a function
f’(z)
g- >
Re
0=0
The cases when
and
generalization
A function f, analytic in E,f(O)=O=
Vk(0)
g in
such that
0
e=0 have been discussed in [6] and [7]
0=0,
respectively.
Definition i.i
Let m
in E with
and m
I
2
be any numbers and S
and S
1
and S (0)=I=S (0).
2
I
SI(0)--0=$2(0)
2
Then
be analytic functions
fgVml
m
if and only if
R
m
I
(s ’(z))
1
f’(z)
m
(S ’(z))
(i.i)
2
2
We have the following special cases.
k+2
k-2
Let
Case A.
k>2_ in (i.I).
---,
ml=
(i)
S
1
---,
are convex univalent functions.
2
Then
the class of functions with bounded boundary rotation if
Vml,m2 =V k
and S
m2=
This was proved by Brannan in
[8].
(ii) V
ml,
m2--TOk(O)
(iii) V
k
if S
coincides with V
mlm2
This resu t is shown in
(iv) V
m I,
S2eT 2(0),
I
and S
2
are close-to-convex univalent func-
[7].
see
tions,
T
T
m2
see
e
and zS’ are a-spiral-like functions.
if zS
I
2
k
[2].
e
if S
and
k
I
e
S2eT 2(0)’
[5]
see
[6] and
Vml,m2
e
T (0) if S and
k
1
-
Case B.
Let S and S be convex univalent functions in (i.i).
2
i
have the following subcases:
e
k-2
-ie
k+2 -ie
If
in (i.i).
e
cose
then feV
(i)
m2=
ml= k+2e cose,
k
-ie
-i
then
cose
(ii) If m
ecos
relation
2.
See [2].
fEVk(O)
----(l-O)e
m2= (l-O)e
I
This is shown in [i].
(i.i).
MAIN RESULTS
Then we
We now proceed to prove the main results for the class V
ver needed,
certatin restrictions on the parameters m
lytic functions S
Theorem
I
and S
2
will be imposed.
2.1
Let f E V
ml,m 2
such that
I
and m
mal’n2
2
in
Whereon
and-
SOME CLASSES OF ANALYTIC FUNCTIONS
497
m
(s ’(z)) I
1
f’(z)
where S
i
and S
2
m
(s’ 2 (z)) 2
Let
are convex univalent in E.
27[
Ix(r)=
0<r<l_
where
and
If’ (re
/
0
(2.1)
dO,
2miX>l; ml,m2>0
Then
(l-r)
lim Sup
r+l
X(r)<A(m l,m 2,X)
I
where
2m2X F (m
A(ml,m2, X)
Proof
1
X
m
o
Is(z)
1
Then
and
I X+
)
(2mlX-l) F(mlX
2
SI’
S ’(z) I>_--------by the distortion theorems for convex functions
2
2
( +rl
-2is subOrdiNate to (l-z)
in E.
Consequently
<_(l+r) 2m2X
I X (r)
[i
I
i0 2m i
1-re
xdO
(l+r)
2m X
2
J2mlX(r),
say
[9]
(2.2)
Now it has been shown by Pommerenke in [I0] that
J (r)
P
r(p-)
1
(l_r)P -I’
2P-IF2 (D)
r(-i
p
+
(p-l)
p>l
r+l
i
)
(2.3)
I
r (p)
(l-r) p-1
using the recurrence and duplication formulae for the Gamma function.
Substitution of
(2.3) in (22) completes the proof.
Corollary 2.1
Let
ml=
k+2
---,
m2=
--.
k-2
Then feV
k
and
lim Sup
r+l
A(k,X)
(k- i) X-I I X(r)<A(k,X)_
(--k-l) I
i
i
2
T1/2(. kX+X-l)r(1/4k,+ X)
(l-r)
where
498
K.I. NOOR AND H. MADIFER
[8].
This result was proved in
Theorem 2.2
Let
SI,S 2
and
fVml,m2
length of the arc f(
zl
given by the formula for z=re
=r)
L (r)---for m
Then
I
,
> 1
m
2
> 0
I
2
zf’
denote the
i@
(z)Ide
0
we have
i
L(r)=0(1)"
(i-r)
2m -i
1
is a constant depending only on m
where 0(i)
Let L(r)
be convex functions.
m
I and 2.
The proof follows immediately from Theorem 2.1 by taking =i.
From Theorem 2.1 and the standard inequality [9,p.ll]
la n <en
i
(i-
i
Y)’
we have the following.
Theorem 2.3
Let feV
in
and be given by (i.I)with f(z)=z+
ml,m 2
-
(2.1) are convex, m
lim Sup n
2-2m
n+
ml=--,k+2
lla n
[n
n+
where S
i
and
Then for n>2
2F(ml+
1
in Theorem 2.3 then feV
and
k
i
-k F (k+ i)
4
1
1/2(k)r (_k4 + )
1
k la
n
I<=-- (2m l-l)F(ml
--k-2
m2
i-
lim Sup
2m
e2
Corollary 2.2
If
1
i> ,m 2 >0
z
e 2
I] <
n
This result was proved in
[8].
Theorem 2.4
Let feV
f(z)
z
mm I
+
m
with S
2
7. a z n
n
n= 2
I
and S
C(ml,m 2)
Let f be given by
ml>l, m2>0.
Then for n>l
lan+ll-I a n
where
convex and
2
<c(ml,m 2 )n
is a constant depending only on m
I
and m
2"
Proof
ZleE and n>l,_ we have
i
(n+l)Zlan+l-nanl= 2r n+l
For
2[
I
0
2
2r
"
o
I.-ZlllZf’(z)lde,
m
IS I
Z-zll Is’
2
z=r
ie
1
m
(z) 2
de
(2.4)
SOME CLASSES OF ANALYTIC FUNCTIONS
It is known [9]
499
that for convex univalent functions S 2
1
’(z)
2
Also, by a result of Golusin [Ii],
zl
such that for all z,
Using
(2.5) and (2.6),
l(z)
zS
1
with
Iz11=r
=r
,
where
ZlEE
there exists a
IZ-Zll ISl(Z) l_<
,
S
(2 5)
l> (l+r)
IS2
2 r
2
(2.6)
l-r
(z) is univalent
(2.4) becomes
2m
(l+r)
(n+l)Zlan+l-na n I< 2r n-i
2m2-i
rn-3
2
2r
m-i
2
[S’(z)[
1
2
l_r
< (l+r)
(l-r)
1
d
2mi-2
where we have used subordination for the function S I.
we obtain the required result.
Putting
=r, r=
-,
Zll
Corollary 2.3
Taking
ml=
k-2
k+2
m2=-q-, k>2’--
and
k
depending only on k.
we obtain fV
.llan+ll-lan II<C(k)--
n
k
-2
2
where C(k)
is a constant
Now we give the radius of convexity problem for the class V
where the functions S
I
and S
2
are in V
ml,m 2
k.
Theorem 2.5
Let
such that
fgV
ml,m 2
m
(s ’(z))
1
f’(z)
where
Sl,S2eVk
ml,m2>0
and
(s 2 (z)) m2
and real.
Then f is convex for
zl
< r where
r is the least positive root of
k
k
[l+m2(l- )]-k(ml+m2)r+[2ml-m2(l+ )-l]r
2
(2. 7)
0
Proof
From definition it easily follows that
(zf’ (z))_’
f’(z)
(zSI (z))
ml
S ’(z)
1
(zS ’(z))’
Now, for S 1 eV k it is known [12] that
(zS l(z))’
> l-kr+r
Re
2
’(z)
1-r
S1
2
m2 s ’(z)
2
+ (l-m I +
m
2)
(2.8)
500
-
K.I. NOOR AND H. MADIFER
Also, by the Paatero representation theorem [3] we have, for
(zS ’(z))’
k+ 2
2
s 2’(z)
so
that
h
I
k- 2
(z)
-----’h2
’4
(z)
(zs ’(z))’
2
(zS ’(z))’
2
Re
<
$2’(z)
--<
i=1,2, and
0
hi(0)
i
k l+r
1 r
<2
s 2’(z)
Thus, using (2.8) and (2.9)
(zf’ (z))
Re
f’(z)
h.1 (z) _>
Re
S2sV k,
we have
k
k
-) ]-k(ml+m2)r+[2ml-m2(l+ )-l]r
[l+m2(l
l-r
2
and this gives us the required result.
Corollary 2.4
If k=2, then
(2.7) reduces to
C, the class of convex functions and equation
SI,S2EV 2
1-2(ml+m 2)r+(2ml-2m2-1)r
and in this case if m
equation (2.7)
k+2
I
k-2
4
m2
4
then
2
=0
Vml,m 2
reduces to V
k
and
reduces to the known result
l-kr+r
which was given in[
2
0
12].
Corollary 2.5
If m
I
>0,
m2--0,
then f is convex for
Izl<
r, where r is the least
positive root of
2
l-kr + (2a- l)r
0
This result has been proved in [13].
Theorem 2.6
Let feV
ml,m 2
such that
m
(s ’(z))
1
f’(z)
m
(s ’(z)) 2
2
SI,S2eV k ml,m2_>0, ml-m2<_l.
Then feVk,, where k’
{ml(k-2)+m2(k+2)+2
and
From the above result, we deduce the following:
(i)
If
Sl,S2eV2
then
feV4m2+
2
and in this case if m
we have the well known result
(ii) If m
I
e,
m2=0
[8] that feV k"
0<_a_<l, then
feVa(k_2)+2.
k+2
I
k-2
SOME CLASSES OF ANALYTIC FUNCTIONS
501
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