Internat. J. Math. & Math. Sci.
VOL. 21 NO. 2 (1998) 369-374
369
ANGULAR ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS
NAK EUN CHO, IN HWA KIM and JI A KIM
Department of Applied Mathematics
Pukyong National University
Pusan 608-737, KOREA
(Received April 1, 1996 and in revised form September 30, 1996)
ABSTRACT. The object of the present paper is to derive some argument properties of certain integral
operators. Our results contain some interesting corollaries as the special cases.
KEY WORDS AND PHRASES: Argument, integral operators, starlike functions, Bazilevi6 functions.
1991 AMS SUBJECT CLASSHICATION CODES: 30C45.
1.
INTRODUCTION
Let A denote the class of functions of the form
z+
f(z)
-
oz
Iz
which are analytic in the open unit disk U { z
< 1}. If f and g are analytic in U, we say that f is
subordinate to g, written f g, if there exists a Schwarz function w(z) in U such that f(z) g(w(z)).
A function f e A is said to be in the class S* [E, F] if
zf’(z) l+Ez
-<
f()
+ FZ (z6U,-I<F<E<I)
The class S*[E,F] was studied in [1,2]. In particular, S’[1 2c, 1] S’(c)(0 _< c < 1) is the well
known class of starlike functions of order a. We observe [2] that a function f is in S"[E,F] if and only
if
f(z)
1-.F < 1-.F
and
Re zf’(z)
}
j(z)’ >
E
1
2
(z 6 U,F
1).
(1.3)
A function f e A is said to be in the class B(p, a, ) if it satisfies
Re{
Zf’(z)f"-I
g,,()
}
>/(z e u)
for some #(# > 0), (0 _< < I) and g e S*(o). Furthermore, we denote BI(/, a,/) bythe subclass
of B(/, a, ,O) for g(z) =_ z e S* (a). The classes B(, a,/) and B (/, a, ) are the subclasses of
Bazilevi6 functions in U [3]. We also note that B(I, a,/) C(a,/) is an important subclass of closeto-convex functions [4].
For a positive real number # > 0 and a function f A, we define the integral operator Jc., by
370
N.E. CliO, I. H KIM AND J. A KIM
Jcu(f)
c+#
tc-l fu(t)dt
;(c >
(1 4)
Kumar and Shukla [5] showed that the integral operator Jc,u(f) defined by (1.4) belongs to the class
S*[E,F] for c >_ u(g-:l)l_F, whenever f S’[E,F]. The operator Je.1, when c N {1,2,3, .}, was
introduced by Bemardi [6]. Further, the operator J. was studied earlier by Libera [7] and
Livingston [8].
In the present paper, we give some argument properties of the integral operator defined by
(1.4). We also generalize the previous results of Libera [7], Owa and Srivastava [9] and Owa and
Obradovi6 10].
MAIN RESULTS
In proving our main results, we shall need the following lemmas.
LEMMA 1 ([11]). Let M(z) and N(z) be regular in U with M(0) N(0) 0, and let/ be real.
If N(z) maps U onto a (possibly many-sheeted) region which is starlike with respect to the origin, then
2.
Re
N’(z)
> (z e U)
= Re
M(z)
< ,O(z U)
= Re
N(’z)
and
Re
N.t{z)
LEMMA 2 ([12]). Let p(z) be analytic in U, p(0)
exists a point zo U such that
Iv()l <
fo
< (z U).
1, p(z) :/:. 0 in U and suppose that there
I1 < Iol
and
where/ > 0. Then we have
o’(o)
ik,
where
whvn argp(zo)
r
2
and
a+-a
k<-1(1)
when argp(zo)
2
where
p(o)
+ m( > o).
With the help of Lemma and Lemma 2, we now derive
THEOREM 1. Let c and # be real numbers with c > 0,
.fA If
> 0 and
1
< F < E < 1 and let
-
ANGULAR ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS
S" [E, F], then
for some g
<
3 71
T(0_< <1,0<_<1)
where arc,. is the imegral operator defined by (1.4) and r/(O < r/_< 1) is the solution of the equation
6
{2(rsin(1-t(E,F)))
"
-1
7+-Tan
7r
c + I+F
+rlcos ’(1
r]
when
_
2
tc(E,F)
PROOF. Let us put
where
M(z)
8in_
7r
zf, (z)
E- F
(2.1)
(2.2)
M(z)
N(z)’
tC-tg(t)dt
t-If"(t)dt t1
c
1
for F= -1,
( c(1-F2)+I-EF
p(z)-
1
for F :/:
tc(E,F))
and
N(z)
Then p(z) is analytic in Uwith p(0)
1.
M’(z)
N’(z)
tc- gu(t)dt.
#
By a simple calculation, we have
N(z)
p(z) 1 +
zp’(z))
zN’(z) p(z)
1(zf’(z)f"-(z) )
-
Since g e S*[E,F], J.(g) e S*[E,F] [5] and hence N(z) is (possibly many-sheeted) starlike function
with respect to the origin. Therefore, from our assumption and Lemma 1, p(z) 0 in U.
If there exists a point zo 6 U such that
[argv(z)[<
and
then, from Lemma 2, we have
for
Izl < Izol
0p’ (o)
where
when arg p(zo
-
372
N.E. CHO, I. H. KIM AND J. AK/M
and
when argp(zo)
2
where
p(zo)
Since Jc.z(g) E
i.(. > 0).
S’[E,F], from (1.2) and (1.3), we have
g’() (&,(g))’
/c
N()
J,.()
-
where
pe’T,
I+E
< C+
-tc(E,F)<<t(E,F) forF# -I,
c+
1-E
<
I .F P
when t,(E, F) is given by (2.2), and
1-E
c+ 2 <p< oo,
--1<<1 for F=
At first, suppose that p(zo)
z0f’(z0)f"-I (z0)
ia(a > 0). For the case F
_/)
-1.
:
1, we obtain
(1 )M’(zo)
N’(zo)
arg
(
1
argp(zo) + arg 1 + z(y.,.(g))’
-
zo/g(z,o)
+ c p(zo)
&(g)
+ (pe’ ], -’ irlk
r 7- + ( )1
_>
r?-"
arg
/ Tan_
_r_,
2
where
have
t(E,F) and 6
g
+ Tan -1
+o g(- )
(
c/
sin (1- tc(E,F))
+cos ’(1. t(E,F))
+EI+F
are given by (2.2) and (2.1), respectively. Similarly, for the case
F
1, we
(f()F-() )
These are a contradiction to the assumption of our theorem.
Next, suppose that p(zo)
ia(a > 0). For the case F
above, we have
arg(zf’(z)fz-(z)
<
"Tr
Tan_
(
4=
1, applying the same method as the
nsin(1-t(E,F))
-[-
I+EI+F /7]C08 "(1
where t(E,F) and 6 are given by (2.2) and (2.1), respectively and for the case F
1, we have
ANGULAR ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS
arg
/’()/"- ()
g"(zo)
373
) ’
B <
2
which are contradictions to the assumption. Therefore we complete the proof of our theorem.
1 in Theorem 1, we have
Taking E 1 2a(0 < a < 1) and F
COROLLARY 1. Let c _> 0, # > 0 and f E A. If
arg
z’f’(z)f"-(z)
<
g(z)
(0 <
< 1, 0 < < 1)
for some g E S* (a), then
arg
where dc,u is the imegral operator defined by (1.4).
REMARK 1. For 6 1, Corollary is the result obtained by Owa and Obradovi6 [10].
1,
1, 6 1 and g(z) z in Theorem l, we have
Setting E 1, F
COROLLARY 2. Let c _> 0 and f A. If
.Re ft(z) >/(0 <_/5 < I),
then
e (s,. (I))’ >/,
where Jc,1 is the imegral operator defined by (1.4).
Letting # 1 in Theorem 1, we have
COROLLARY 3. Let c _> 0 and 1 _< F < E
arg
g(z)
< 1 and let f A. If
)1 <-(0_</<1,0<6<1)
/
for some g 6 S" [E, F], then
where Jc,1 is the integral operator defined by (1.4) and r/(0 < r/< 1) is the solution of the equation (2.1).
1 in Corollary 3, we have
Taking E 1 2a(0 _< a < 1) and F
A.
>_
If
COROLLARY 4. Let c 0 and f
then
arg
Jc, l(f)
cz
<
-,
where Jc,1 is the integral operator defined by (1.4).
1 and//= 1 in Corollary 3 and Corollary 4, we obtain the
Putting E 1 2a(0 _< a < 1), F
following result of Owa and Srivastava [9].
COROLLARY 5. If the function f defined by (1.1) is in the class C(c,/), then the integral
operator Jc,1 (f)(c > 0) defined by (1.4) is also in the class c(a,/).
REMARK 2. Taking c =/ 0 and c 1 in Corollary 5, we obtain the result given earlier by
Libera [7]
374
N.E. CliO, I. H. KIM AND J. AKIM
-
By using the same technique as in proving Theorem 1, we have
THEOREM 2. Let c and # be real numbers with c > 0,/ > 0 and
fA.
If
,qV(z)
<
for some g E S* [E, F], then
1
< F < E < 1 and let
(/ > 1, 0 < 6 <_ 1)
where Jc,, is the integral operator defined by (1.4) and r/(0 < r/< 1) is the solution of the equation (2.1)
Putting g 1 2a(0 < a < 1), F
1, # 1 and 6 1 in Theorem 2, we have the following
result by Owa and Srivastava [9].
COROLLARY 6. Let c > 0 and .f E A. If
()
for some g
<(>1)
S* (c), then
z(Jc,1 (f))’
}
where ,Jc,1 is the integral operator defined by (1.4).
ACKNOWLEDGEMENT. The authors would like to thank Professor M. Nunokawa for his thought
encouragement and much valuable advice in the preparation of this paper. This work was partially
supported by Non Directed Research Fund, Korea Research Foundation, 1996 and the Basic Science
Research Program, Ministry of Education, Project No. BSRI-96-1440.
[l]
[2]
[3]
[4]
[5]
[6]
REFERENCES
JANOWSKI, W., Some extremal problems for certain families of analytic functions, Bull. de
L’Acad Pol. des ScL 21 (1973), 17-25.
SILVERMAN, H. and SILVIA, E.M, Subclasses of starlike functions subordinate to convex
functions, Can. d. Math. 37 (1985), 48-61
SINGH, R., On Bazilevi functions, Proc. Amer. Math. Soc. l (1952), 169,-815.
KAPLAN, W., Close-to-convex Schlicht functions, Michigan Math. 1 (1952), 169-185.
KUMAR, V. and SHUKLA, S.L., On p-valent starlike functions with reference to the Bemardi
integral operator, Bull. Austral. Math. Soc. 30 (1984), 37-43.
BERNARDI, S.D., Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969),
429-446.
[7] LIBERA, R.J., Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965),
755-758
[8] LIVINGSTIN, A.E., On the radius of univalence of certain analytic functions, Proc. Amer. Math.
Soc. 17 (1966), 352-357.
[9] OWA, S. and SRIVASTAVA, H.M., Some applications of the generalized Libera imegral operator,
Proc. Japan Acad 62, Ser. A (1986), 125-128.
[10] OWA, S. and OBRADOVI(, M., Certain subclasses of Bazilevi/: functions of type c, Internat. d.
Math. Math. ScL 9 (1986), 347-359.
11] MILLER, S.S. and MOCANU, P.T., Second order differemial inequalities in the complex plane, o
Math. Anal. AppL 65 (1978), 289-305.
12] NUNOKAWA, M., On the order of strongly starlikeness of strongly convex functions, Proc. Japan
Acad 69, Set. A (1993), 234-237.