http://ijmms.hindawi.com © Hindawi Publishing Corp.
CERTAIN INTEGRAL OPERATOR AND STRONGLY
STARLIKE FUNCTIONS
JIN-LIN LIU
Received 15 June 2001
LetS∗(ρ,γ)denote the class of strongly starlike functions of orderρand typeγand let C(ρ,γ)be the class of strongly convex functions of orderρand typeγ. By making use of an integral operator defined by Jung et al. (1993), we introduce two novel families of strongly starlike functionsSβα(ρ,γ)andCβα(ρ,γ). Some properties of these classes are discussed.
2000 Mathematics Subject Classification: 30C45, 30C75.
1. Introduction. LetAdenote the class of functions of the form
f (z)=z+ ∞
n=2
anzn (1.1)
which are analytic in the unit discE= {z:|z|<1}. A functionf (z)belonging toAis said to be starlike of orderγif it satisfies
Re
zf(z) f (z)
> γ (z∈E) (1.2)
for someγ (0≤γ <1). We denote byS∗(γ)the subclass ofAconsisting of functions which are starlike of orderγinE. Also, a functionf (z)inAis said to be convex of orderγif it satisfieszf(z)∈S∗(γ), or
Re
1+zf(z)
f(z)
> γ (z∈E) (1.3)
for someγ (0≤γ <1). We denote byC(γ)the subclass ofAconsisting of all functions which are convex of orderγinE.
Iff (z)∈Asatisfies
arg
zf(z) f (z) −γ
<π2ρ (z∈E) (1.4)
for someγ (0≤γ <1)andρ (0< ρ≤1), thenf (z)is said to be strongly starlike of orderρand typeγinE, and denoted byf (z)∈S∗(ρ,γ). Iff (z)∈Asatisfies
arg
1+zf(z)
f(z) −γ
for someγ (0≤γ <1)andρ (0< ρ≤1), then we say thatf (z)is strongly convex of orderρand typeγinE, and we denote byC(ρ,γ)the class of such functions. It is clear thatf (z)∈Abelongs toC(ρ,γ)if and only ifzf(z)∈S∗(ρ,γ). Also, we note thatS∗(1,γ)=S∗(γ)andC(1,γ)=C(γ).
Forc >−1 andf (z)∈A, we recall the generalized Bernardi-Libera-Livingston inte-gral operatorLc(f )as
Lc(f )=c+1
zc
z
0t
c−1
f (t)dt. (1.6)
The operatorLc(f )whenc∈N= {1,2,3,...}was studied by Bernardi [1]. Forc=1, L1(f )was investigated by Libera [4].
Recently, Jung et al. [2] introduced the following one-parameter family of integral operators:
Qαβf (z)=
α+β β
α zβ
z
0
1−t z
α−1
tβ−1f (t)dt (α >0, β >−1, f∈A). (1.7)
They showed that
Qα
βf (z)=z+
∞
n=2
Γ(β+n)Γ(α+β+1) Γ(β+α+n)Γ(β+1)anz
n, (1.8)
whereΓ(x)is the familiar Gamma function. Some properties of this operator have been studied (see [2,3]). From (1.7) and (1.8), one can see that
zQαβ+1f (z)
=
(α+β+1)Qαβf (z)−(α+β)Qα+
1
β f (z). (1.9)
It should be remarked in passing that the operatorQα
β is related rather closely to the Beta or Euler transformation.
Using the operatorQαβ, we now introduce the following classes:
Sβα(ρ,γ)=
f (z)∈A:Qαβf (z)∈S∗(ρ,γ), zQα
βf (z)
Qβαf (z)
≠γ∀z∈E
,
Cβα(ρ,γ)=
f (z)∈A:Qαβf (z)∈C(ρ,γ),
zQαβf (z)
Qαβf (z)
≠γ∀z∈E
.
(1.10)
It is obvious thatf (z)∈Cβα(ρ,γ)if and only ifzf(z)∈Sβα(ρ,γ).
In this note, we investigate some properties of the classesSβα(ρ,γ)andCβα(ρ,γ). The basic tool for our investigation is the following lemma which is due to Nunokawa [5].
Lemma1.1. Let a functionp(z)=1+c1z+c2z2+···be analytic inEandp(z)≠0
(z∈E). If there exists a pointz0∈Esuch that
argp(z)<π 2ρ
|z|<z0, argpz0=π
then
z0pz0
pz0
=ikρ, (1.12)
where
k≥12
a+1
a when argp
z0
=π2ρ
,
k≤ −12
a+1
a whenargp
z0
= −π 2ρ
,
(1.13)
andp(z0)1/ρ= ±ia (a >0).
2. Main results. Our first inclusion theorem is stated as follows.
Theorem2.1. The classSβα(ρ,γ)⊂Sβα+1(ρ,γ) forα >0,β >−1,0≤γ <1and α+β≥ −γ.
Proof. Letf (z)∈Sα
β(ρ,γ). Then we set
zQαβ+1f (z)
Qα+1
β f (z)
=(1−γ)p(z)+γ, (2.1)
wherep(z)=1+c1z+c2z2+···is analytic inEandp(z)≠0 for allz∈E. Using (1.9) and (2.1), we have
(α+β+1) Q α βf (z) Qαβ+1f (z)
=(α+β+γ)+(1−γ)p(z). (2.2)
Differentiating both sides of (2.2) logarithmically, it follows from (2.1) that
zQαβf (z)
Qαβf (z)
−γ=(1−γ)p(z)+ (1−γ)zp(z)
(α+β+γ)+(1−γ)p(z). (2.3)
Suppose that there exists a pointz0∈Esuch that
argp(z)<π2ρ |z|<z0, argp
z0=π
2ρ. (2.4)
Then, by applying Lemma 1.1, we can write that z0p(z0)/p(z0) = ikρ and that
(p(z0))1/ρ= ±ia (a >0).
Therefore, if argp(z0)= −(π/2)ρ, then
z0Qαβfz0
Qα βf
z0 −γ=(
1−γ)pz0
1+ z0p
z0
/pz0
(α+β+γ)+(1−γ)pz0
=(1−γ)aρe−iπρ/2
1+ ikρ
(α+β+γ)+(1−γ)aρe−iπρ/2
From (2.5) we have
arg
z0
Qα βf
z0
Qαβf
z0
−γ
= −π2ρ+arg
1+ ikρ
(α+β+γ)+(1−γ)aρe−iπρ/2
= −π2ρ+tan−1
kρ
(α+β+γ)+(1−γ)aρcosπρ 2
×
(α+β+γ)2+2(α+β+γ)(1−γ)aρcosπρ 2
+(1−γ)2a2ρ−kρ(1−γ)aρsinπρ 2
−1
≤ −π2ρ,
(2.6)
wherek≤ −(1/2)(a+1/a)≤ −1,α+β≥ −γ, which contradicts the conditionf (z)∈
Sβα(ρ,γ).
Similarly, if argp(z0)=(π/2)ρ, then we have
arg
z0Qαβfz0
Qα βf
z0
−γ
≥π2ρ, (2.7)
which also contradicts the hypothesis thatf (z)∈Sβα(ρ,γ).
Thus the functionp(z)has to satisfy|argp(z)|< (π/2)ρ (z∈E), which leads us to the following:
arg
zQα+1
β f (z)
Qαβ+1f (z)
−γ
<π2ρ (z∈E). (2.8)
This evidently completes the proof ofTheorem 2.1.
We next state the following theorem.
Theorem2.2. The classCα
β(ρ,γ)⊂Cβα+1(ρ,γ)forα >0,β >−1,0≤γ <1, and α+β≥ −γ.
Proof. By definition (1.10), we have
f (z)∈Cβα(ρ,γ)⇐⇒Qαβf (z)∈C(ρ,γ)⇐⇒z
Qαβf (z)
∈
S∗(ρ,γ)
⇐⇒Qα β
zf(z)∈S∗(ρ,γ)⇐⇒zf(z)∈Sα β(ρ,γ)
⇒zf(z)∈Sα+1
β (ρ,γ)⇐⇒Qαβ+1
zf(z)∈S∗(ρ,γ)
⇐⇒zQβα+1f (z)
∈S∗(ρ,γ)⇐⇒Qαβ+1f (z)∈C(ρ,γ)
⇐⇒f (z)∈Cβα+1(ρ,γ).
(2.9)
Theorem2.3. Letc >−γand0≤γ <1. Iff (z)∈Aandz(QαβLcf (z))/QαβLcf (z) ≠γfor allz∈E, thenf (z)∈Sβα(ρ,γ)implies thatLc(f )∈Sβα(ρ,γ).
Proof. Letf (z)∈Sα
β(ρ,γ). Put
zQα βLcf (z)
Qα βLcf (z)
=γ+(1−γ)p(z), (2.10)
wherep(z)is analytic inE,p(0)=1 andp(z)≠0(z∈E). From (1.6) we have
zQα βLcf (z)
=
(c+1)Qα
βf (z)−cQαβLcf (z). (2.11)
Using (2.10) and (2.11), we get
(c+1) Q α βf (z) QαβLcf (z)
=(c+γ)+(1−γ)p(z). (2.12)
Differentiating both sides of (2.12) logarithmically, we obtain
zQα βf (z)
Qα βf (z)
−γ=(1−γ)p(z)+ (1−γ)zp(z)
(c+γ)+(1−γ)p(z). (2.13)
Suppose that there exists a pointz0∈Esuch that
argp(z)<π 2ρ
|z|<z0, argpz0=π
2ρ. (2.14)
Then, applyingLemma 1.1, we can write thatz0p(z0)/p(z0)=ikρand(p(z0))1/ρ= ±ia (a >0).
If argp(z0)=(π/2)ρ, then
z0
Qαβf
z0
Qα βf
z0 −γ=(
1−γ)pz0
1+ z0p
z0
/pz0
(c+γ)+(1−γ)pz0
=(1−γ)aρeiπρ/2
1+ ikρ
(c+γ)+(1−γ)aρeiπρ/2
.
(2.15)
This shows that
arg
z0Qαβfz0
Qα βf
z0 −γ
=π 2ρ+arg
1+ ikρ
(c+γ)+(1−γ)aρeiπρ/2
=π 2ρ+tan
−1kρ(c+γ)+(1−γ)aρcosπρ 2
×
(c+γ)2+2(c+γ)(1−γ)aρcosπρ 2
+(1−γ)2a2ρ+kρ(1−γ)aρsinπρ 2
−1
≥π2ρ,
(2.16)
Similarly, we can prove the case argp(z0)= −(π/2)ρ. Thus we conclude that the functionp(z)has to satisfy|argp(z)|< (π/2)ρfor allz∈E. This shows that
arg
zQαβLcf (z)
QαβLcf (z)
−γ<π
2ρ (z∈E). (2.17)
The proof is complete.
Theorem 2.4. Let c >−γ and 0≤ γ < 1. If f (z)∈ A and (z(Qα
βLcf (z)))/ (QαβLcf (z))≠γfor allz∈E, thenf (z)∈Cβα(ρ,γ)implies thatLc(f )∈Cβα(ρ,γ).
Proof. Using the same method as inTheorem 2.2we have
f (z)∈Cβα(ρ,γ)⇐⇒zf(z)∈Sβα(ρ,γ) ⇒Lc
zf(z)∈Sβα(ρ,γ)
⇐⇒zLcf (z)∈Sα
β(ρ,γ)⇐⇒Lcf (z)∈Cβα(ρ,γ).
(2.18)
References
[1] S. D. Bernardi,Convex and starlike univalent functions, Trans. Amer. Math. Soc.135(1969), 429–446.
[2] I. B. Jung, Y. C. Kim, and H. M. Srivastava,The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl.176 (1993), no. 1, 138–147.
[3] J. L. Li,Some properties of two integral operators, Soochow J. Math.25(1999), no. 1, 91–96. [4] R. J. Libera,Some classes of regular univalent functions, Proc. Amer. Math. Soc.16(1965),
755–758.
[5] M. Nunokawa,On properties of non-Carathéodory functions, Proc. Japan Acad. Ser. A Math. Sci.68(1992), no. 6, 152–153.
