243
Internat. J. Math. & Math. Sci.
VOL. 20 NO. 2 (1997) 243-248
SUBCLASSES OF UNIVALENT FUNCTIONS SUBORDINATE
TO CONVEX FUNCTIONS
YONG CHAN KIM
IL BaNG JUNG
Department of Mathematics
Department of Mathematics
eUngnam University
Kyungpook National University
Taegti*102-701, KOREA
ongsan 712-749, KOREA
(Received May 17, 1995 and in revised form August 30, 1995)
ABSTRACT. In this paper, we define a new subclass Ao(A, B) of univalent functions and investigate
several interesting characterization theorems involving a general class S" [A, B] of starlike functions
KEY WORDS AND PHRASES: Univalent function, subordination, a-convex function
1991 AMS SUBJECT CLASSIFICATION CODES: 30C45, 30D30
1.
INTRODUCTION AND DEFINITIONS
Let .A denote the class of functions normalized by
z+
f(z)
anzn,
(I I)
which are analytic in the open unit disk ( { z z E C and Izl < I }. Further, let S denote the class of
all functions in L/which are univalent in L/
A function f(z) belonging to ,.q is said to be starlike of order a (0 _< a < 1) if and only if
We denote by S* (a) the subclass of S consisting of functions which are starlike of order a
A function f(z) belonging to 8 is said to be convex of order a (0 < a < 1) if and only if
z.f."(z)
We denote by/C(a) the subclass of 8 consisting of functions which are convex of order a We note that
(14)
S*(a) C_ S* (O) _= S* (O_<o< I)
and
c() c_ c(0)
c (0 _< < 1).
()
With a view to introducing an interesting family of analytic functions, we should recall the concept of
subordination between analytic functions Given two functions f(z) and g(z), which are analytic in L,
the function f(z) is said to be subordinate to g(z) if there exists a function h(z), analytic in L/with
(0)
0 and
I()I < l,
( 6)
such that
()
We denote this subordination by
(()) ( s u).
(.7)
24/,
Y C KIM AND B JUNG
f(z) -< ().
(1 s)
In particular, if g(z) is univalent in H, the subordination (1 8) is equivalent to
g(O) and f(/,t’)C .q(U).
f(O)
(1 9)
Janowski
introduced the class 7:’[A, B] For 1 < B < A < 1, a function p, analytic in/g with
1, belongs to the class T’[A, B] ifp(z) is subordinate to (1 + Az)/(1 + Bz) Also S’[A,B] and
L:[A, B] denote the subclasses of $ consisting of all functions f(z) such that
p(0)
,y’(,)
(,f’(z))’
E 79[A,B] and
f(z)
fl(z)
79[A,B],
(1 10)
respectively. We note that S*[- 1, 1] ,5" and K:[- 1,1] K:
DEFINITION 1. Let cz be a real number. A function f(z) belonging to the class .4 with
(f(z)/z)f’(z) 0 is said to be a-convex inL/if and only if
Re
(1-a)I’()+a
f()
1+
(1 11)
>0.
f’()
Also we denote the class of a-convex functions by .Ma. Then it is easy to see that
A4a
{f
S" Re
I(1-a) zf’(z)f(z)
>0,
zL/1.
(1 12)
See Eenigenberg and Miller [5] for further information on them
We now define the class .Ado(A, B) as follows: If a is a real number, then
A4o(A,B)=(fS’((1-a)
+a 1+
zf"(z)
"]
f’(z) ]
"<
-I <_B < A <_
1
+ Az
I + Bz’
I,zLt}.
(1 13)
Clearly, we have
.Ada (1,- 1) =.Ma, M (A, B)
K:[A, B],
(1 14)
and
S’[A,B].
.Mo(A,B)
2.
(1 15)
MAIN RESULTS
Applying the method of the integral representation [2] for functions in .Mo(A, B) (a > 0), it is not
difficult to deduce
LEMMA 1. The function f(z) is in
belonging to the class S" [A, B] such that
f(z)
PROOF. Setting g(z)
JVlo(A,B),
a
> 0, if and only if there exists a function g(z)
{g(t)}l/t-ldt
,
f(z) {zf’(z)/f(z)} so that (2.1) is satisfied, we observe that
zg’(z)
zft(z) a
1+ zft’(z)h
(l-a)
g(z)
f(z) +
f’(z) )
(
Hence f e Jvlo (A,B) ifand only ifg S’[A,B].
Before stating our first theorem, we need the following definition
DEFINITION 2. Let c be a complex number such that Re c > 0, and let
(2 1)
245
SUBCLASSES OF UNIVALENT FUNCTIONS
N
[[c (1 + 2Rec)
N(c)
1/=
+
2Nz/(1- z2) and b
If h is the univalent function h(z)
(ef [3]) function Q as
Imc]/Rec.
h-l(c),
hI(z + b)/(l +-z)],
Qc(z)
-
zE
(22)
then we define the "open door"
L.
(23)
TItEOREM 1. Let f E .h4 (A, B) (a > 0), and let
l+bz
S"
PROOF. Since f
Then f
a2_(z).
(24)
.Mo (A, B) (a > 0), it follows that there exists a function g S*[A, B]
such
that
1
I(*)
{g(t)}l/’t-ldt
(25)
by using Lemma 1. By the hypothesis, we also have
1
a
zg’(z)
g(z)
-<
1
a
+ Bz
-<
(z).
(26)
Thus, by a result of Miller and Mocanu ([3], Corollary 3.1), we have
{g(t)}l/t-ldt e S’.
f(z)
LEMMA 2. (Mocanu [4]) Let 7:’ be an analytic function in/d satisfying 7 --< Qc If p is analytic in
L, p(0) I/c, and
zp’(z) + P(z)p(z)
1,
(2 7)
then Re p(z) > 0 in H
Making use of Lemma 2, we now prove
TI]EOREM 2. Let f E .M, (A, B) (a
> 0), and let
zff (z) f(z)
f(z)
Then f C S* [A, B].
PROOF. If we set p(z)
zf’(z)/f(z), then
P’()
p(z)
v(z)
(28)
f"(z)
1
f, (--;-
(29)
Hence
(I-a)
Since f
zff(z) +a 1+
f(z)
f"(z))
--p(z)+a
(2 10)
.M (A, B),
p(z) +
Setting T’(z)
p(z) + lip(z)
zff (z)
p(z)
1 + Az
1 + Bz
(2 11)
1, we obtain
v’ ()
(2 12)
246
Y C KIMAND B JUNG
and P --<
by the hypothesis (2.8)
Thus, by Lemma 2, we have
Rep(z) > 0 (z e U).
Since a
(2 13)
> O,
Re{1-P(z)}
>Oc
(zE/d).
(2 14)
Also (1 + Az)/(1 + Bz) (with
1 < B < A < 1) is a convex univalent function
appealing to a known result ([6], Theorem 7), we conclude from (2 11) and (2 14) that
p(z) -.<
l+Az
1
Therefore, by
(2 15)
+ Bz"
This evidently completes the proof of Theorem 2
As an example of ([7], Corollary 3.2, see also [9]), consider the case when a
1 < B < A < 1, and A # B. Then the differential equation
q(z) + a
zq’(z)
q(z)
1
1
+ Az
+ Bz
> 0,
(2 16)
has a univalent solution given by
if
B:0
q()
(2 17)
if B=0.
If p(z) is analytic in H and satisfies
p(z) +
zp’ (z)
p(z)
then
-
p(z) -< q(z) -<
+ Az
+ Bz
(2 18)
1 +Az
l+Bz"
(2.19)
1
1
Hence, by the equations (2.11) and (2 19), we obtain
TItEOREM 3. Let a > 0 and f E .Mo (A, B). Then
zf’ (z)
f(z)
where q(z) is given by (2.17).
TItEOREM 4. K:(o) C .Ma(1
PROOF. If we define
ho(z)
2a,
1
’< q(z) -,<
1 + az
1 + Bz
1)(0 <
< 1).
c
+ (1 2c)z (O<a<l)
1-z
(2.20)
(221)
then we can easily see that f 6/C(a) if and only if
+ zf"()
?,()
ho()
(cf 10], Equation (9)). Hence, by Theorem of 10], we have
(2 22)
SUBCLASSES OF UNIVALENT FUNCTIONS
(2 23)
f()
Therefore we conclude from [8, Lemma 2.2] that
f(z)
This completes the proof of Theorem 4
+c 1 +
2/, 7
f’(z) ]
-
ha(z).
(2 24)
ACKNOWLEDGMENT. This work was partially supported by KOSEF (project No 94-1400-02-01-3)
and TGRC-KOSEF, and by the Basic Science Research Institute Program (BSRI-95-1401)
REFERENCES
[1 JANOWSKI, W, Some extremal problems for certain families of analytic functions I, Ann. Polon.
Math. 25 (1973), 297-326
[2] MOCANU, P T., Une propri6t6 de convexit6 g6n6ralise6 darts la th6orie de la repr6sentation
conforme, Mathemattca (Cluj) 11 (34)(1969), 127-133.
[3] MILLER, S.S. and MOCANU, P.T., Classes of univalent integral operators, J. Math. Anal. Appl.
157 (1991), 147-165
[41 MOCANU, P.T., Some integral operators and starlike functions, Rev. Roumatne Math. Pures Appl.
31 (1986), 231-235.
[5] EENIGENBURG, P.J. and MILLER, S.S., The H’ classes for c-convex functions, Proc. Amer.
Math. Soc. 38 (1973), 558-562.
[6] MILLER, S.S and MOCANU, P.T., Differential subordinations and univalent functions, Mchgan
Math. ,I. 28 (1981), 157-171
[7] MILLER, S.S. and MOCANU, P T., Univalent solutions of Briot-Bouquet differential equations,
J. Dfferential Equation 56 (1985), 297-309.
[8] NOOR, K.I., On some univalent integral operators, J. Math. Anal. Appl. 128 (1987), 586-592
[9] OWA, S. and SRIVASTAVA, H.M., Analytic solutions of a class of Briot-Bouquet differential
equations, in Current Topics m Analyttc Function Theory (H.M Srivastava and S Owa, Editors),
252-259, World Science Publishing Company, Singapore, New Jersey, London, and Hong Kong,
1992.
[10] SALAGEAN, G S., Subclasses of univalent functions, Complete Analysis: Fifth Romaman-Fimsh
Seminer, Part (C Andreian Cazacu, N. Boboc, M Jurchescu, and Siciu, Editors), 362-372,
Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983