Internat. J. Math. & Math. Sci.
VOL. 15 NO. 3 (1992) 517-522
517
A CLASS OF UNIVALENT FUNCTIONS WITH VARYING ARGUMENTS
K.S. PADMANABHAN and M. JAYAMALA
The Ramanuan Institute,
University of Madzas,
Madras
600 005,
and
Department of Mathematics,
Queen Mary’ s College,
Madras
600 005,
India.
(Received April 24, 1990 and in revised form August 7, 1991)
ABSTRACT.
f(z)
am zm
z + Z
is said to be in
V(6n) if the
analytic and univalent function f in the unit disc E is nozmallsed
by
f(O)
O, f’(O)
I and arg a n
exists a real number
Vn(A,B)
6
<
and
On+(n-l) p
union of V(en,)
The
and all possible real number
consists of functions f
Dn+If(z)
Dnf (z)
-I
< B <_
Iw(z)l <
A
for all n.
such that
V((n,).
said to be in
sequence
p
(n
x(mod 2=) then f is
taken over all possible
is denoted by V.
V such that
1+Aw(z)
I+BW (Z)
l, where n E N U
1, z g E.
If further there
0}
and
w(z) is analytic, w(O)
0
In this paper we find the coefficient
inequalities, and prove distortion theorems.
KEY WORDS AND PHRASES.
Varying arguments, Ruscheweyh derivative,
Distortion theorems, Coefficient estimates.
kTHEATICS SUBJECT CLASSIFICATION No. 3OC45
INTRODUCTION.
Let A denote the class of functions f(z) analytic in the
unit dsc E
Let S denote the subclass of A
{z zl <
X
consisting functions normaltsed by f(O)
are univalent
f(z)
n E. e
az= and
I tch
Hadamard pruct (feg)(z) of two functions
g(z)
(-) (z)
O and f’ (O)
Z
z
m
bmz=
in A Is given by,
b=zm.
K.S. PADMANABHAN and M. JAYAMALA
518
Let
Dnf(z)
[2] observed
Ruscheweyh
[0
* f(z), n E N U
(l_z)n+l
that
where N
z(zn-lf(z))(n)/n!.
Dnf(z)
th Ruscheweyh derivative of
is called the n
(Stlverman
DEFINITIC 1.
f(z)
[3]).
en
to be in V(en) if f E S and ag a n
overall possible sequences
If futhe
fo all n.
[e
a
and
ea
possible
numbe
V.
is denoCed by
Now we define the class
V such that
Dn+lf(z)
phi(z)
and
Let
Kn(A,B) denote
E Vn(A,B).
w(z)
Vn(A,B)
consisting of functions
-l
=’
tO}
that zf’(z)
is said
en+(n-l)p (= 2x),
V(en,). e union of V(en,) taken
then f Is said to be in
z E E.
amzm
z+ Z
such at
there exists a real nber
N U
f(z) by
Dnf(z)
AI-AmSri Ill.
is analytic,
<
w(O)
A
<
< I vee
B
I*(=)1 < z,
O and
the class of functions f
V such
2. COEFFZCIENT INEQUALITIES.
V.
Let f
HECEM I.
(n+m.l):
(m-z)’
On+l)!
ms2
E
PROOF.
Vn (A, B).
Suppose f
l+B(z)
Dnf(z)
is analytic,
Since Re
w{O)
O and
w(z) < [w(z)l < 1,
n+l== 1= =’
;" .-.
)2
"
Since f
c=lami
if and only if
< (B-A).
(B+Z)(n)-(I+A)(n+I).
whe.e Cm
w(z)
Vn(A,B)
Then f
V, f
eaZ nbe
-
Then
<- A <
I*(=)1 <
<_
1
,=
)-(n)jamzm-
V(%,)
]a===_.’-"
fo ,=e ,equenc,
such that
em+(m-z)p
E.
get
obatn on =tmpZtftcaon,
(nz)
(n+l)) (=’1)1 [a(n)-A(n+l)
Zie, in
B
= =(rood 2).
t
= eP.
< z.
[
(2.2)
and
519
CLASS OF UNIVALENT FUNCTIONS WITH VARYING ARGUMENTS
Then we get,
(n+m-l).
(n+l)-(n)
]lamlzm-Ae
i(em +"-I )
m2 (ni)(mL1).,
’-"m2 (n+l)-(m-i)l [B(n+m)-A(n+l)]lamlz
e
e
<.
(n+m-I)
(n+1mLi)"--.-[(n+m)-(n+l)]aml
Z
m=2
< (B-A)- Z
m=2
E
m=2
(2.3)
rm-1
(n+m-)’
[B(nm)-A(n+l)][a
-(+1.)(mi’l),
mz
(ne.l)’
(a+1)I(m-[) [(B+)(n+m)-(1)(n+l)]laml rm-i <
m-1
(B)
Hence,
Z
m=2
)
(n+(i)n+m-I(m-l)’
CmJam] rm-1
< (B-A).
(2.4)
Letting r-> 1 we get (2,1).
V and satisfies (2.1). In view of
(2.4) which is implied by (2.1), since rm-I < 1, we have,
Conversely, suppose f
which gives
(2.2) and hence follows that f
COROLLARY 1.
If f
V is in
Vn(A,B )
Vn(A,B).
then,
(n+l) (m-) (B-A)
(n+m-i’): Cm
The equality holds o the unction
[’am’ <
m _) 2.
(m-l) (B-A)
z + (n+l);
(n+m-Z).’
f(z)
THEOREM 2.
(A,B)
c,.
Let f E V.
Then f(z)
given by,
elem zm
z
z+ Z
am zm
m=2
if and only if
Z
(n+m-l)
(n/i): (m-i):
mClm
am
< (B-A)
e.
is in
K.S. PADMANABHAN and M. JAYAMALA
520
Let f(z)
HEOREM 3.
f(z)
f g
(m 2).
[m+(m-l)]
where
m
2,3,..., z
.,
1.
f f(z)
PR.
mf(z)
(n+m-1) Cmm
arg am
th
,=e-
E.
mfm(z)
ZO, then,
(n+,) (m-,)
(nm-1) cm
(nl) ’mi)
m=2
--V"(A’B)’ with
fl(z) z and
if and only if f can be expressed as f(z)
O and
Z
g
m
C’
m
(n-l)
Vn(A,B)
m
flne
(B-A)e
z +
where
amzm
z+ Z
z m(S)
(S).
(i-.,) (s-A)
V (A B)
Hence f
Convezsely
f(z)
am zm g Vn(A,B)
[amlCm
’n+l)(ml)(B)
m" om eozem
mfm(Z) pmZ+amzm,
I
1
(n-l)
m
define,
z + Z
Z
mfm(Z)
Z
THEOREM 4.
fro(z)
Then f
f(z)
Define
z +
2,3,...
m=2
Pm
1,
m=2
Since,
m
z +
am zm
fl(z)
z and
1 and o
(n+l) (m-l) :_(,-A) zm
1 Z
O.
m -’2,3,... z
-(n+m-t)! mCm
Kn(A,B ) if and onZy tf
Pmfm(Z) whez’e pro_> O
m:l
and define
E.
f can be exp.---essed as
and
]:
IZa -1.
3. DISTORTION THEOREMS.
THEOREM 5.
n(A,B).
class V
Let; the function f(z)
a_zm be in the
m2
Then,
IzI-(B-A)Izl2/C2 <_.
z+ E
If(z)l <_.
Izl+(B..-)lzl2/c2
t-2(B-A)Izl/C2 ! It"()l <_ Z*2(B-)I=I/C2.
Poo,
I’(=)1
z
Iz+m:2
aBz’I <_ I=1+1=12m=2z I%1
(3.1)
(3.2)
CLASS OF UNIVALENT FUNCTIONS WITH VARYING ARGUMENTS
and
Izl-ll 2 z fatal. Since-(+.l)!(m.l)
and f(z)
function of m
(A.). by
If(z)[ >
inceasln9
521
is an
Theorem I.
we have
(n+l-(nJ)[) C2mZ=2 I%1 < ===Z (n+)(..) %1%1
< ()
that is,
z ,1%1 < B-A
(3.3)
m=2
(3.3) we get (3.1)
m=2
m=2
and
I’()1
Since
z- I1
mCm
’(n+l’)!’ (+’1):
<
-1%1.
is an increasing function of m
(+i)!-ml:-
(n+m-l)
z
(n+m-l)
mCm
z
(n:+1) !" 2 m=2
=1%1
!
2 and
by Theorem 1, we have,
(nl):! m!-
(n+l)! C2
>_
=
m=2
(n+-):
cml%l <-
"(n+’l) ’(-l)’!’
(B-A)
that is,
z
m=2
=1%1
From (3.4) we get (3.2).
-2
Further for the function f(z)
z (B’A) z 2
%
we can see that the results of the Theorem are sharp.
Let f(z)
COROLLARY 2.
z+ Z
amzm
be in the class
Vn(A,B).
Then f(z) is included in a disc with its center at the origin
and radius r given by r
(C2+B-A)/C2
and f’(z) is included
in a disc with its center at the origin and radius r given
I
by r I
]/C2.
[C2*2(B-A)
THEGREM 6.
Let the function f(z)
z+
Izl-(B-)lzl2/2
C2
Z-(B-A)Izl/C2
I,’(z)l <_ Z+(B-)Izl/C2
! If(z)l
amzm
IzI+(B-A)Izl2/2
be in the
C2
and
for z g E.
The results are sharp for the function f(z)
z+(B-A)z2/2
C2.
K.S. PADMANABHAN and M. JAYAMALA
522
REFERENCES
1. H.$.AI-Ami:i, On Ruscheweyh derivatives,
..
(zgeo),
ev-.
.Ann...PoLon.
2. S.Ruscheweyh, New ctte_ta for untvaZent functions,
Math. Soc.,49 (1975), 109-115.
,
9.
3. H.Stlvean, UnlvaZent Functions dth varying Arguments,
No.2 (1981).
Houston JounaZ of Math.