ANGULAR ESTIMATIONS OF CERTAIN ANALYTIC FUNCTIONS
NAK EUN CHO AND JI A KIM
ABSTRACT. In the present paper, we investigate some argument properties of certain analytic functions and the integral preserving properties in a sector. Our results include several previous results as special cases.
1. Introduction
Let A denote the class of functions of the form
(1.1)
00
fez) =z+L anzn
n=2
which are analytic in the open unit disk U = {z : Izl < 1}. Iff and 9 are analytic in U, we say that 9 is subordinate to f, written 9 -< f
or g(z) -< fez), if f is univalent in U, g(O) = f(O) and g(U) ~ feU).
A function f ofA issaid to be in the class8*(a), the class of starlike functions of order a, if
Re{zJ~~~)}>a (0<a ~ 1, z EU).
The class 8* of starlike functions is identified by 8*(0) = 8*. A func- tion f E A is said to be in the class 8*(m,M) if
zf'(z)
I fez) - ml <M (z E U, Im - 11 <M ~ m).
Received October 26, 1996.
1991 Mathematics Subject Classification: 30C45.
Key words and phrases: argument, starlike function, integral operator, close-to- covex function.
This paper was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1996.
The classS*(m,M) was introduced by Jakubowski[3]. It is clear that m> ! andS(m, M) c S*(m - M) c S*.
A function j E A is said to be in the class C(o:,(3) if there is a starlike function9 of order 0:such that
Re{z:~;;)}> {3 (0 ~(3 < 1, Z E U).
Kaplan[4] proved that every j E C(O,O), the class of close-to-convex functions, is univalent. Also, C(0:,(3) provides an interesting general- ization of the class of close-to-convex functions[13].
Many authors[1,7,8] have studied the integral operators of the form
(1.2) fcU) = c+1 rtc - 1j(t)dt,
ZC la
where c is a suitably chosen real constant and j belongs to some favoured classes of univalent functions. In particular, Kumar and Shukla[6] showed that the integral operator fcU) defined by (1.2) maps S(m,M) into itself for c~ -(m - M).
In the present paper, we give some argument properties of certain analytic functions and the integral operator defined by (1.2). We also generalize the previous results of Bulboadi[2] , Libera[7], Owa and Sri- vastava[ll] and Sakaguchi[12].
2. Main results
In proving our main results, we shall need the following lemmas.
LEMMA 1 ([9]). Leth E K., the classof convex functions in U and letA(Z) be analyticinU with ReA(z) ~O. Ifp(z) isanalyticin U and p(O) = h(O), then
p(Z)+A(Z)Zp'(z) -< h(z) (z E U) implies
p(Z) -<h(z) (z E U).
LEMMA 2 ([10]). Let p(z) be analyticin U, p(O) = 1, p(z) -1=0 in U. Suppose that there exists a pointZo E U such that
larg p(z)! < 7r: for Izl < Izol
and
larg p(zo)I
where/3 >O. Then wehave
zop'(zo) p(zo) where
- 2'7r/3
- ik/3,
and
where
k ~ ~(a +~) when arg p(zo) = 7r:
k ~ -~(a+~) when arg p(zo) = _ 7r:
1 •
p(zo)iI = ±za (a> 0).
LEMMA 3 ([5,6]). The function f of the form (1.1) belongs to S(m, M) ifand only if there exists a function w regular in U which satisfies w(O) = 0, Iw(z)1 < 1 for z EU and
(2.1) zj'(z) = 1+Aw(z) (z EU), fez) 1-Bw(z)
where A = (M2 - m2+m)/M and B = (m -l)/M.
With the help of Lemma 1 and Lemma 2, we now derive
THEOREM 1. Letf E A and9 E S*(m, M). H
larg ((1-1')~i;i +1'~:i;i -13)1 < ~{j (-y2:0, 0~ 13<1, 0<15~1),
then
I (f(z) )I 7r'fJ
arg g(z) - /3 < 2'
where 'fJ(0 < 'fJ ~ 1)is the solution of the equation
2 ( /''fJsin2!:.(l- 1Sin-1M) )
(2.2) 6='fJ+-Tan -1 2 7r m .
7r m +M +/''fJcos~(l- ~Sin-l ~)
Proof Let us put
1 (f(z) )
p(z) = 1-f3 g(z) - f3 .
Thenp(z) is analytic inU withp(O) = 1. By a simple calculation, we have
1 (f'(z») g(z),
1 _ f3 g'(z) - f3 = p(z)+ zg'(z) zp (z).
Therefore we obtain
(l_'Y)f(z) +'Yf'(z) -f3=(l-f3)(P(z)+ 'Yg(z) zP'(z»).
g(z) g'(z) zg'(z)
Applying the assumption and Lemma 1 with A(Z) ='Yg(z)jzg'(z), we see that Rep(z) > 0 in U and hence p(z) =1= 0 in U. Ifthere exists a point Zo E U such that
larg p(z)I < 1r'
2'fJ for Izl < Izol
and
larg p(zo)1 -
then, from Lemma 2, we have zop'(zo}
p(zo) - ik'fJ, where
and
where
p(zo);j1 = ±ia (a> 0).
Since 9 E S*(m,M), we have
zg'(z) iJ!!2.
g(z) =pe 2 ,
where
{
m-M < p < m+M
-~Sin-lM < A,. < ~Sin-lM .
1r m 0/ 1r m
At first, suppose that p(zo)";j1 = ia(a >0). Then we obtain
arg ((1- ')')f(zo) +')'f'(zo) - {3)
g(zo) g'(zo)
( ( ,),g(zo) zp'(zo)))
= arg (1 - {3)p(zo) 1+zg'(zo) p(zo) ( ')'g(zo) zop'(ZO))
= arg p(zo)+arg 1+ '() ()
zog Zo P Zo
7r1] . ( -J!.£ 1 )
= T +arg 1+')'(pet 2 ) - i1]k
7r1] -l( ')'1]ksin~(l-4» )
= -+Tan
2 p+')'1]kcos~(l-4»
7r1] (')'1]Sin1I.(l- ~Sin-lM) )
> _ +Tan-1 2 1r m
- 2 m +M +')'1]cos~(1- ~Sin-l ~)
7r
= "2<5,
where <5 is given by (2.2). This is a contradiction to the assumption of our theorem.
Next, suppose that p(zo)Ti1 = -ia (a > 0). Applying the same method as the above, we have
(f'(zo) ) 7r'l'} -l( 'Y'I'}sin~(1-~Sin-l~) ) arg - - - (3 < - - - Tan
g'(zo) - 2 m+M +'Y'I'}oos2[(1- '£Sin-2 1M )
1r m
=--6,7r
2
where <5 is given by (2.2), which contradicts the assumption. This
completes the proof of our theorem. 0
Let us choose m =N - a(N - 1) and M =(1-a), where N ~ 1 and 0 ~ a < 1. Then Im - 11 < M ~ m, A = alN+(1 - 2a) and
B = 1 - 1/Nin Lemma 3. Now as N ---? 00,A ---? 1 - 2a and B ---? l.
Inthis case, the relation (2.1) reduces to
z!,(z) _ 1+(1-2a)w(z) (z E U),
fez) 1-w(z)
which is a necessary and sufficient condition forf to be in8*(a). Hence we have the following
COROLLARY 1. Letf E A and9 E 8*(a). H
then
I (f(z) )I 1rb
arg g(z) - {3 < 2'
REMARK 1. (i) Putting a = 0 and b= 1 in Theorem 1, we obtain the result of Bulboadi[2].
(ll) For the case a ={3= 0 and 'Y= b=1, Corollary 1 is the result by Sakaguchi[12].
Taking m = 1, M ---? 0, "( = 1, (3 =0 and g(z) = z in Theorem 1, we have
COROLLARY 2. Letf E A. H
larg j'(z)I < 1rb2 (0<b~ 1),
then
larg f~Z)I < 2'1r17
where17 (0<17 ::; 1)is the solution of the equation
By using the same technique in the proof of Theorem 1, we have
THEOREM 2. Letf EA and 9 ES*(m, M). If
then
Iarg ((J - g(z)f(z»)1 < 1fT,2'
where77(0 < 77$ 1) is the solution of the equation (2.2).
Letting m =M, m --+00 and 8 =1 in Theorem 2, we have
COROLLARY 3. Letf E A and 9 E S*. If
{ fez) j'(z)}
Re (1 - 'Y)g(z) +'Yg'(z) < (J C'Y ~ 0, (J> 1), then
{ fez) } Re g(z) < (J.
Next, we prove
THEOREM 3. Let c be a real number with c ~ 0 andlet f E A. If
larg (z:~~;) -(J)I < ~8 (0 $ (J <1, 0 <8 $ 1)
for some 9 ES* (m, M), then
wherele is the integral operator deEned by (1.2) and77(0 < 77 $ 1) is the solution of the equation
2 77sinJI..(l- ~Sin-l(--.M..-))
(2.3) 8= 77+ -Tan-1( 2 7r e+m ).
7r C+m +M +77COS~(1- ~Sin-lC.ttm))
Proof Putting
where
T(z) p(z) = S(z) ,
and
S(z) =1Z tC-1g(t)dt,
we see thatp(z)isanalytic inUwithp(O) = 1. By a simple calculation, we have
T'(z) S(z),
S'(z) = p(z)+ zS'(z) zp (z)
= _l_(Z!'(Z) _(3).
1-[3 g(z)
Since 9 E S(m,M), Ic(g) E S(m, M) [5] and hence S(z) is (possibly many-sheeted) starlike function with respect to the orign. Therefore, from our assumption and Lemma 1, p(z) #: 0 in U. Since Ic(g) E
S(m, M), we have
zS'(z) z (Ic(g))' i1!.2
S(z) = Ic(g) +c=pe 2 , where
{
c+m-M < p < c+m+M,
-£Sin-1(.M-)1 I " c + m < A,.'Y < £Sin-1(.M-).11" c+m
The remaining part of the proof is similar to that of Theorem 1 and so
we omit it. 0
Taking m =N - a(N -l),M =N(l- a)(O $ a < l),N ~ 00
and 8 =1 in Theorem 3, we obtain the following result of Owa and Srivastava[ll].
COROLLARY 4. H the function f defined by (1.1) is in the class C(a,[3), then the integral operator Ic(f) (c ~0) defined by(1.2)isalso in the clasSC(a,(3).
REMARK 2. Taking 0:=/3=0 and c=1 in Corollary 4, we obtain the result given earlier by Libera[7].
By using the same technique as in proving Theorem 3, we have
THEOREM 4. Letc be a realnumberwith c ~ 0 andlet f E A. H
Iarg (/3 - Zgf('z(z)) )I < 27r{j (/3> 1, 0<{j ~ 1) forsome 9 E8(m,M), then
larg (/3- Z(le(J))'(J))!
le(g)
7rTJ
< 2'
wherele is the integral operator defined by (1.2) andTJ(O <TJ ~ 1) is the solution ofthe equation (2.3).
Putting m = N - o:(N - 1),M = N(l - 0:)(0 ~ 0: < 1),N --+ 00
and {j = 1 in Theorem 4, we have the following result by Owa and Srivastava[ll] .
COROLLARY 5. Letc ~ 0 andf E A. H Re {zj'(z)} < /3 (/3 > 1)
g(z) for some 9 E8*(0:), then
wherele is the integral operator defined by(1.2).
ACKNOWLEDGEMENT. The authors would like to thank the referee for his helpful advices.
References
[1] S. D. Bernardi, Convex and starlike univalent junctions, Trans. Amer. Math.
Soc. 135 (1969),429-446.
[2] T.Bulboaca, Differential subordinationbyconvex functions, Mathematica(Cluj) 29 (1987), 105-113.
[3] Z. J. Jakubowski, On the coefficients of starlike functions of some classes,Ann.
Polon. Math. 26 (1972), 305-313.
[4] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185.
[5] V.Kumar and S. L. Shukla, Certain integrals for classes of p-valent meromor- phic functions, Bull. Austral. Math. Soc. 25 (1982),85-97.
[6] V. Kumar and S. L. Shukla, Jakubowski integral operators, J. Austral. Math.
Soc. (series A) 37 (1984), 117-127.
[7] R. J. Libera, Some classes of regular univalent !unctions, Proc. Amer. Math.
Soc. 16 (1965), 755-758.
[8] A. E. Livingstin, On the radius of univalence of certain analytic functions, ibid. 17 (1966), 352-357.
[9] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent func- tions, Michigan Math. J. 28 (1981), 157-171.
[10] M. Nunokawa, On the order of strongly starlikeness of strongly convex !unc- tions, Proc. Japan Acad. 69, Ser. A(1993), 234-237.
[11] S. Owa and H. M. Srivastava, Some applications of the generalized Libera in- tegral operator,Proc. Japan Acad. 62 (1986), 125-128.
[12]K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72-75.
[13] H. Silverman, On a class of close-to-convex functions, Proc. Amer. Math. Soc.
36 (1972), 477-484.
Department of Applied Mathematics Pukyong National University
Puasn 608-737, Korea
