ON SOME CLASSES OF MEROMORPHIC FUNCTIONS DEFINED BY A MULTIPLIER TRANSFORMATION
Alina Totoi
Abstract. Forp∈N∗
let Σp denote the class of meromorphic functions of the
formg(z) = a−p
zp +a0+a1z+· · ·, z∈U , a−˙ p 6= 0. In the present paper we introduce
some new subclasses of the class Σp, denoted by ΣSp,λn (α) and ΣSp,λn (α, δ), which
are defined by a multiplier transformation, and we study some properties of these subclasses.
2010Mathematics Subject Classification: 30C45.
Keywords: meromorphic functions, multiplier transformations. 1. Introduction and preliminaries
Let U = {z ∈ C : |z| < 1} be the unit disc in the complex plane, ˙U =U \ {0},
H(U) ={f :U →C:f is holomorphic inU},Z={. . .−1,0,1, . . .},N={0,1,2, . . .} and N∗
=N\ {0}. Forp∈N∗
let Σp denote the class of meromorphic functions in ˙U of the form g(z) = a−p
zp +a0+a1z+· · ·, z∈U , a−˙ p 6= 0.
We will also use the following notations: Σp,0 ={g∈Σp :a−p= 1},
Σ∗ p(α) =
g∈Σp : Re
−zg
′
(z)
g(z)
> α, z ∈U
, where α < p, Σ∗
p(α, δ) =
g∈Σp :α <Re
−zg
′
(z)
g(z)
< δ, z ∈U
, where α < p < δ, H[a, n] ={f ∈H(U) :f(z) =a+anzn+an+1zn+1+. . .}for a∈C, n∈N∗,
We know that Σ∗
1(α) is the class of meromorphic starlike functions of order α, when 0≤α <1.
Forn∈Z, p∈N∗
, λ∈C with Reλ > p, let the operator Jn
p,λ on Σp be defined
as
Jp,λn g(z) = a−p
zp + ∞
X
k=0
λ−p k+λ
n
akzk,whereg(z) = a−p
zp + ∞
X
k=0
akzk.
Let p∈N∗
, λ∈C with Reλ > p. We have: 1. J−1
p,λg(z) =
1
λ−pzg ′
(z) + λ
λ−pg(z), g∈Σp,
2. J0
p,λg(z) =g(z), g∈Σp,
3. J1
p,λg(z) = λ−p
zλ
Z z
0
tλ−1
g(t)dt=Jp,λ(g)(z), g∈Σp,
4. If g∈Σp withJp,λn g∈Σp, then Jp,λm (Jp,λn g) =J n+m
p,λ g, form, n∈Z.
Remark 1. Letp∈N∗
andλ∈Cwith Reλ > p. We know from [7] that if g∈Σp, then Jp,λ(g)∈Σp, hence, using item 4 and the induction, we obtain
Jp,λn g∈Σp for all n∈N ∗
.
We notice from item 1 that for g∈Σp we have J −1
p,λg∈Σp, so J−n
p,λg∈Σp for all n∈N ∗
.
Therefore, for n∈Z, p∈N∗
, λ∈Cwith Reλ > p, we have Jn
p,λ: Σp →Σp.
Now it is easy to see that we have the next properties for Jp,λn , when Reλ > p: 1. Jn
p,λ(Jp,λmg(z)) =J n+m
p,λ g(z), n, m∈Z, g∈Σp ,
2. Jp,γn (Jp,λm g(z)) =Jp,λm (Jp,γn g(z)), n, m∈Z, g∈Σp,Reγ > p, 3. Jp,λn (g1+g2)(z) =Jp,λn g1(z) +Jp,λn g2(z), forg1, g2 ∈Σp, n∈Z,
4. Jp,λn (cg)(z) =cJp,λn g(z), c∈C∗
, n∈Z, 5. Jp,λn (zg′
(z)) =z(Jp,λn g(z))′
= (λ−p)Jn−1
Remark 2. 1. When λ= 2 and p= 1, we have
J1n,2g(z) =
a−1
z +
∞
X
k=0
(k+ 2)−na kzk,
and this operator was studied by Cho and Kim [1] forn∈Zand by Uralegaddi and Somanatha [8] forn <0.
2. We also have
z2J1n,2g(z) =Dn(z 2
g(z)), g∈Σ1,0,
where Dn is the well-known Sˇalˇagean differential operator of order n [5], de-fined by Dnf(z) =z+
∞
X
k=2
knakzk, f(z) =z+
∞
X
k=2
akzk . 3. Jn
p,λ is an extension to the meromorphic functions of the operatorKpn, defined
on A(p) = (
f ∈H(U) :f(z) =zp+
∞
X
n=1
ap+nzp+n )
, introduced in [6]. Also, forn≥0 we find that Kpn is the Komatu linear operator, defined in [2]. 4. It is easy to see that for n > 0, Jp,λn is an integral operator while J−n
p,λ is a
differential operator with the property J−n
p,λ(Jp,λn g(z)) =g(z).
Lemma 1. [7] Let n∈N∗
, α, β ∈R, γ ∈C with Reγ−αβ >0. If P ∈H[P(0), n] with P(0)∈Rand P(0)> α, then we have
Re
P(z) + zP
′
(z)
γ−βP(z)
> α⇒ReP(z)> α, z∈U.
Definition 1. [3, pg. 46], [4, pg. 228] Let c ∈C with Rec > 0 and n∈ N∗
. We consider
Cn=Cn(c) = n
Rec
" |c|
r
1 +2Rec
n + Imc
#
.
If the univalent function R:U →C is given by R(z) = 2Cnz
1−z2, then we denote by Rc,n the ”Open Door” function, defined as
Rc,n(z) =R z+b
1 + ¯bz
!
= 2Cn (z+b)(1 + ¯bz)
Theorem 1. [7] Let p ∈ N∗
, Φ, ϕ ∈ H[1, p] with Φ(z)ϕ(z) 6= 0, z ∈ U. Let
α, β, γ, δ ∈ C with β 6= 0, δ +pβ = γ +pα and Re (γ −pβ) > 0. Let g ∈ Σp and suppose that
αzg ′
(z)
g(z) +
zϕ′
(z)
ϕ(z) +δ≺Rδ−pα,p(z). If G=Jp,α,β,γ,δΦ,ϕ (g) is defined by
G(z) =Jp,α,β,γ,δΦ,ϕ (g)(z) =
γ−pβ zγΦ(z)
Z z
0
gα(t)ϕ(t)tδ−1
dt
β1
, (1)
then G∈Σp with zpG(z)6= 0, z∈U,and
Re
βzG ′
(z)
G(z) +
zΦ′
(z) Φ(z) +γ
>0, z∈U.
All powers in (1) are principal ones.
Taking β = α = 1, δ = γ, Φ = ϕ ≡ 1, in the above theorem, and using the notation Jp,γ instead ofJp,1,11,1,γ,γ, we obtain the corollary:
Corollary 1. Letp∈N∗, γ
∈CwithReγ > pand let the functiong∈Σp satisfying
the condition
zg′
(z)
g(z) +γ ≺Rγ−p,p(z), z∈U. Then
G(z) =Jp,γ(g)(z) = γ−p
zγ
Z z
0
g(t)tγ−1
dt∈Σp,
with zpG(z)6= 0, z∈U,and Re
γ+zG
′
(z)
G(z)
>0, z∈U.
2. Main results
Definition 2. For p∈N∗
, n∈Z, λ∈C,Reλ > p andα < p < δ we define
ΣSp,λn (α) =
g∈Σp: Re
−
zJp,λn g(z)
′
Jn p,λg(z)
> α, z ∈U ,
ΣSp,λn (α, δ) =
g∈Σp :α <Re
−
zJp,λn g(z)
′
Jn p,λg(z)
Remark 3. 1. We have g ∈ΣSp,λn (α) if and only if Jp,λn g ∈Σ∗
p(α), respectively g∈ΣSp,λn (α, δ) if and only ifJp,λn g∈Σ∗
p(α, δ).
2. Using the equalityz(Jp,λn g(z))′
= (λ−p)Jn−1
p,λ g(z)−λJp,λn g(z), we can easy see
that for Reλ > pthe condition
α <Re
−
zJp,λn g(z)
′
Jp,λn g(z)
< δ, z∈U,
is equivalent to
Reλ−δ <Re "
(λ−p)J
n−1
p,λ g(z) Jn
p,λg(z)
#
<Reλ−α, z∈U. (2)
3. We have
ΣSp,λ0 (α, δ) = Σ∗ p(α, δ),
ΣSp,λ1 (α, δ) =
g∈Σp :G(z) = λ−p
zλ
Z z
0
tλ−1
g(t)dt∈Σ∗ p(α, δ)
.
The following theorem gives us a connection between the sets ΣSp,λn (α) and ΣSn−1
p,λ (α), respectively between ΣSp,λn (α, δ) and ΣSn −1
p,λ (α, δ).
Theorem 2. Let p∈N∗
, n∈Z, λ∈Cwith Reλ > p, α < p < δ andg∈Σp. Then
g∈ΣSp,λn (α)⇔Jp,λ(g)∈ΣSn −1
p,λ (α),
respectively
g∈ΣSp,λn (α, δ)⇔Jp,λ(g)∈ΣSn −1
p,λ (α, δ),
where Jp,λ(g)(z) = λ−p
zλ
Z z
0
tλ−1
g(t)dt.
Proof . We know that g ∈ ΣSp,λn (α, δ) is equivalent to Jp,λn g ∈ Σ∗
p(α, δ). Since Jp,λn g=Jn−1
p,λ (J
1
p,λg), we haveJ n−1
p,λ (J
1
p,λg)∈Σ ∗
p(α, δ), which is equivalent to Jp,λ1 (g)∈ΣSn−1
Since Jp,λ1 (g) =Jp,λ(g), we obtain Jp,λ(g)∈ΣSn−1
p,λ (α, δ). Therefore, g∈ΣSp,λn (α, δ)⇔Jp,λ(g)∈ΣSn
−1
p,λ (α, δ).
The proof for the first equivalence is similar, so we omit it.
A corollary presented in [7] holds that: Corollary 2. [7] Letp∈N∗
, γ ∈Cand α < p < δ≤Reγ. Ifg∈Σ∗
p(α, δ), then G=Jp,γ(g)∈Σ
∗ p(α, δ).
Theorem 3. Let p ∈ N∗
, n ∈ Z, λ, γ ∈ C with Reλ > p and α < p < δ ≤ Reγ. Then
g∈ΣSp,λn (α, δ)⇒Jp,γ(g)∈ΣSp,λn (α, δ).
Proof . Becauseg∈ΣSp,λn (α, δ) we haveJp,λn (g)∈Σ∗
p(α, δ), hence, from Corollary 2,
we obtain
Jp,γ(Jp,λn (g))∈Σ∗ p(α, δ).
Using the fact that Jp,γ1 (Jp,λn (g)) =Jp,λn (Jp,γ1 (g)), whereJp,γ1 (g) =Jp,γ(g), we obtain
Jp,λn (Jp,γ(g))∈Σ∗p(α, δ),
which is equivalent toJp,γ(g)∈ΣSp,λn (α, δ).
Corollary 3. Let n∈Z, p∈N∗
, λ∈C and α < p < δ≤Reλ. Then we have ΣSp,λn (α, δ)⊂ΣSp,λn+1(α, δ).
A theorem presented in [7] holds that: Theorem 4. [7] Let p∈N∗
, β >0, γ∈C andα < p < Reγ
β ≤δ.
If g∈Σ∗
p(α, δ), with
βzg ′
(z)
g(z) +γ ≺Rγ−pβ,p(z), z∈U, then G=Jp,β,γ(g)∈Σ∗p(α, δ).
Using this theorem, forβ = 1, we get the next result.
Theorem 5. Let n∈Z, p∈ N∗
, λ, γ∈C with Reλ > p and α < p <Reγ ≤δ. If
g∈ΣSp,λn (α, δ) and satisfies the condition
zhJn p,λ(g)(z)
i′
Jp,λn (g)(z) +γ ≺Rγ−p,p(z), z∈U, then Jp,γ(g)∈ΣSp,λn (α, δ).
We omit the proof because it is similar with that of Theorem 3. If we consider in Theorem 5 thatδ → ∞ we get:
Theorem 6. Let n ∈ Z, p ∈ N∗
, λ, γ ∈ C with Reλ > p and α < p < Reγ. If
g∈ΣSp,λn (α) and satisfies the condition
zhJp,λn (g)(z)i
′
Jn p,λ(g)(z)
+γ ≺Rγ−p,p(z), z∈U,
then Jp,γ(g)∈ΣSp,λn (α).
Theorem 7. Let n ∈ N, p ∈ N∗
, λ, γ ∈ C and α < p < Reγ ≤ Reλ ≤ δ. If
h∈ΣSp,λn (α, δ) and satisfies the condition
zh′
(z)
Proof . Let us denote H =Jp,γ(h).Because the conditions of Corollary 1 are full-filled, we have H∈Σp withzpH(z)6= 0, z∈U,and
Re
zH′
(z)
H(z) +γ
>0, z∈U. (3)
Since Reγ ≤Reλ, we obtain from (3) that Re
zH′
(z)
H(z) +λ
>0, z∈U.
Using now Corollary 1, we have Jp,λH ∈Σp with zpJp,λH(z)6= 0, z∈U, and
Re z
(Jp,λH)′(z) Jp,λH(z)
+λ
>0, z∈U.
By induction, we obtainzpJp,λn H(z)6= 0, z∈U, and
Re "
z(Jn p,λH)
′
(z)
Jn p,λH(z)
+λ
#
>0, z∈U. (4)
Since Reλ≤δ, we get from (4) that
Re "
−z(J
n p,λH)
′
(z)
Jp,λn H(z) #
< δ, z ∈U. (5)
From the definition ofH we have
γH(z) +zH′
(z) = (γ−p)h(z), z∈U .˙ (6) We apply the operator Jp,λn to (6) and after using the properties of Jp,λn , we obtain
(γ−p)Jp,λn h(z) =γJp,λn H(z) +z(Jp,λn H(z))′
. (7)
Fromh∈ΣSp,λn (α, δ), we have
α <Re "
−z(J
n p,λh(z))
′
Jn p,λh(z)
#
< δ,
which is equivalent to (see Remark 3, item 2)
Reλ−δ <Re "
(λ−p)J
n−1
p,λ h(z) Jp,λn h(z)
#
From (7), we have
(λ−p)J
n−1
p,λ h(z) Jn
p,λh(z)
= (λ−p)γJ
n−1
p,λ H(z) +z(J n−1
p,λ H(z)) ′
γJn
p,λH(z) +z(Jp,λn H(z))
′ . (9)
Let us denoteP(z) =−z(J
n
p,λH(z)) ′
Jp,λn H(z) .Using the fact that
z(Jp,λn H(z))′
= (λ−p)Jn−1
p,λ H(z)−λJ n p,λH(z),
we obtain
P(z) = (p−λ)J
n−1
p,λ H(z)
Jp,λn H(z) +λ, z∈U. Hence
Jn−1
p,λ H(z) Jp,λn H(z) =
P(z)−λ
p−λ . (10)
If we apply the logarithmic differential to (10), we get
z(Jn−1
p,λ H(z)) ′
Jn−1
p,λ H(z)
−z(J
n
p,λH(z)) ′
Jn p,λH(z)
= zP
′
(z)
P(z)−λ,
so,
z(Jn−1
p,λ H(z)) ′
Jn−1
p,λ H(z)
=−P(z) + zP
′
(z)
P(z)−λ. (11)
Using (10) and (11) we obtain from (9),
(λ−p)J
n−1
p,λ h(z) Jn
p,λh(z)
= (λ−p)J
n−1
p,λ H(z) Jn
p,λH(z)
·
γ+z(J
n−1
p,λ H(z)) ′
Jn−1
p,λ H(z) γ+z(J
n
p,λH(z)) ′
Jp,λn H(z) =
= (λ−p)P(z)−λ
p−λ ·
γ−P(z) + zP
′
(z)
P(z)−λ
γ−P(z) =λ−P(z) +
zP′
(z)
P(z)−γ. (12)
From (8) and (12), we get Reλ−δ <Re
λ−P(z) + zP
′
(z)
P(z)−γ
which is equivalent to
α <Re
P(z) + zP
′
(z)
γ−P(z)
< δ. (13)
Because we know from (5) that ReP(z) < δ, z ∈ U, we have only to verify that ReP(z)> α, z ∈U.To show this we will use Lemma 1.
We know thatJp,λn H ∈Σp and since we have proved thatzpJp,λn H(z)6= 0, z∈U,
we get thatP is analytic inU. We also have Reγ−α >0 andP(0) =p > α. Since the hypothesis of Lemma 1 is fulfilled for β = 1, we obtain ReP(z) > α, z ∈ U, which is equivalent to
Re "
−z(J
n
p,λH(z)) ′
Jp,λn H(z) #
> α, z∈U. (14)
Since H ∈Σp, we get from (5) and (14) thatH ∈ΣSp,λn (α, δ).
If we considerγ =λ, in the above theorem, we obtain:
Corollary 4. Let n∈N, p∈N∗
, λ∈Candh∈ΣSn
p,λ(α, δ)withα < p <Reλ≤δ.
If
zh′
(z)
h(z) +λ≺Rλ−p,p(z), z∈U. Then Jp,λ(h)∈ΣSp,λn (α, δ).
Takingn= 0 in Corollary 4, we get:
Corollary 5. Let p∈N∗
, λ∈C and α < p <Reλ≤δ. Ifh∈Σ∗
p(α, δ) with zh′
(z)
h(z) +λ≺Rλ−p,p(z), z∈U, then Jp,λ(h)∈Σ∗p(α, δ).
Corollary 6. Let p∈N∗
, λ∈C and α < p <Reλ. If h∈Σ∗
p(α) with zh′
(z)
h(z) +λ≺Rλ−p,p(z), z∈U, then Jp,λ(h)∈Σ∗p(α).
We remark that Corollary 5 and Corollary 6 were also obtained in [7].
Theorem 8. Let n ∈ Z, p ∈ N∗, λ, γ
∈ C with Reλ > p and α < p < Reγ. If
g∈ΣSp,λn (α) withJp,γ(Jp,λn (g)(z))6= 0, z∈U˙, then
Jp,γ(g)∈ΣSp,λn (α).
Proof . Let G = Jp,γ(g). We know from Remark 1 that if g ∈ Σp and Reγ > p,
then G∈Σp.
Let
P(z) =−z(J
n p,λG(z))
′
Jp,λn G(z) , z∈U.
We have Jp,λn (G) =Jp,λn (Jp,γ(g)) =Jp,γ(Jp,λn (g))6= 0 ˆın ˙U. So P ∈H(U).
Making some calculations, similar to those from the proof of Theorem 7, we get: −z(J
n p,λg(z))
′
Jn p,λg(z)
=P(z) + zP
′
(z)
γ−P(z), z∈U. Fromg∈ΣSp,λn (α) we have
Re "
−z(J
n p,λg(z))
′
Jn p,λg(z)
#
> α, z ∈U,
which is equivalent to Re
P(z) + zP
′
(z)
γ−P(z)
> α, z∈U.
Since P ∈H(U) with P(0) = p > α and Reγ > α, we have from Lemma 1 (when
β = 1) that ReP(z)> α, z ∈U, hence
Taking n= 0 andλ=γ in the above theorem we get the next corollary, which was also obtained in [7].
Corollary 7. Let p∈N∗
, γ ∈C and α < p <Reγ. If g∈Σ∗
p(α) with zpJp,γ(g)(z)6= 0, z∈U, then Jp,γ(g)∈Σ∗p(α).
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Alina Totoi
