Paper3-Acta25-2011. 127KB Jun 04 2011 12:03:17 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 25/2011
pp. 41-52

ON SOME CLASSES OF MEROMORPHIC FUNCTIONS DEFINED
BY A MULTIPLIER TRANSFORMATION

Alina Totoi

Abstract. For p ∈ N∗ let Σp denote the class of meromorphic functions of the
a−p
form g(z) = p + a0 + a1 z + · · · , z ∈ U̇ , a−p 6= 0. In the present paper we introduce
z
n (α) and ΣS n (α, δ), which
some new subclasses of the class Σp , denoted by ΣSp,λ
p,λ
are defined by a multiplier transformation, and we study some properties of these
subclasses.
2010 Mathematics 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, U̇ = U \ {0},
H(U ) = {f : U → C : f is holomorphic in U }, Z = {. . .−1, 0, 1, . . .}, N = {0, 1, 2, . . .}
and N∗ = N \ {0}.
For p ∈ N∗ let Σp denote the class of meromorphic functions in U̇ of the form
g(z) =

a−p
+ a0 + a1 z + · · · , z ∈ U̇ , a−p 6= 0.
zp

We will also use the following notations:
Σp,0 = {g∈ Σp : a−p =1},


zg ′ (z)
∗
Σp (α) = g ∈ Σp : Re −
> α, z ∈ U , where α < p,

g(z)




zg ′ (z)
∗
Σp (α, δ) = g ∈ Σp : α < Re −
< δ, z ∈ U , where α < p < δ,
g(z)
H[a, n] = {f ∈ H(U ) : f (z) = a + an z n + an+1 z n+1 + . . .} for a ∈ C, n ∈ N∗ ,
An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + an+2 z n+2 + . . .}, n ∈ N∗ , and for n = 1
we denote A1 by A and this set is called the class of analytic functions normalized
at the origin.
41

A. Totoi - Meromorphic functions defined by a multiplier transformation

We know that Σ∗1 (α) is the class of meromorphic starlike functions of order α,
when 0 ≤ α < 1.

n on Σ be defined
For n ∈ Z, p ∈ N∗ , λ ∈ C with Re λ > p, let the operator Jp,λ
p
as

∞ 
∞
a−p X λ − p n
a−p X
k
n
Jp,λ g(z) = p +
ak z , where g(z) = p +
ak z k .
z
k+λ
z
k=0

Let p ∈

N∗ ,

k=0

λ ∈ C with Re λ > p. We have:

−1
1. Jp,λ
g(z) =

λ
1
zg ′ (z) +
g(z), g ∈ Σp ,
λ−p
λ−p

0 g(z) = g(z), g ∈ Σ ,
2. Jp,λ

p
Z z
1 g(z) = λ − p
3. Jp,λ
tλ−1 g(t)dt = Jp,λ (g)(z), g ∈ Σp ,
zλ
0
n g ∈ Σ , then J m (J n g) = J n+m g, for m, n ∈ Z.
4. If g ∈ Σp with Jp,λ
p
p,λ p,λ
p,λ

Remark 1. Let p ∈ N∗ and λ ∈ C with Re λ > p. We know from [7] that if g ∈ Σp ,
then Jp,λ (g) ∈ Σp , hence, using item 4 and the induction, we obtain
n
Jp,λ
g ∈ Σp for all n ∈ N∗ .
−1
We notice from item 1 that for g ∈ Σp we have Jp,λ

g ∈ Σp , so
−n
Jp,λ
g ∈ Σp for all n ∈ N∗ .
n :Σ →Σ .
Therefore, for n ∈ Z, p ∈ N∗ , λ ∈ C with Re λ > p, we have Jp,λ
p
p
n , when Re λ > p :
Now it is easy to see that we have the next properties for Jp,λ
n (J m g(z)) = J n+m g(z), n, m ∈ Z, g ∈ Σ ,
1. Jp,λ
p
p,λ
p,λ
n (J m g(z)) = J m (J n g(z)), n, m ∈ Z, g ∈ Σ , Re γ > p,
2. Jp,γ
p
p,λ
p,λ p,γ

n (g + g )(z) = J n g (z) + J n g (z), for g , g ∈ Σ , n ∈ Z,
3. Jp,λ
1
2
1 2
p
p,λ 1
p,λ 2
n (cg)(z) = cJ n g(z), c ∈ C∗ , n ∈ Z,
4. Jp,λ
p,λ
n (zg ′ (z)) = z(J n g(z))′ = (λ − p)J n−1 g(z) − λJ n g(z), n ∈ Z, g ∈ Σ .
5. Jp,λ
p
p,λ
p,λ
p,λ

42

A. Totoi - Meromorphic functions defined by a multiplier transformation

Remark 2.

1. When λ = 2 and p = 1, we have
∞

n
g(z) =
J1,2

a−1 X
+
(k + 2)−n ak z k ,
z
k=0

and this operator was studied by Cho and Kim [1] for n ∈ Z and by Uralegaddi
and Somanatha [8] for n < 0.
2. We also have

n
z 2 J1,2
g(z) = Dn (z 2 g(z)), g ∈ Σ1,0 ,

where Dn is the well-known Sǎlǎgean differential operator of order n [5], de∞
∞
X
X
n
k
n
k ak z , f (z) = z +
ak z k .
fined by D f (z) = z +
k=2

k=2

n is an extension to the meromorphic functions of the operator K n , defined
3. Jp,λ

p
(
)
∞
X
on A(p) = f ∈ H(U ) : f (z) = z p +
ap+n z p+n , introduced in [6]. Also,
n=1

for n ≥ 0 we find that Kpn is the Komatu linear operator, defined in [2].

n is an integral operator while J −n is a
4. It is easy to see that for n > 0, Jp,λ
p,λ
−n n
differential operator with the property Jp,λ
(Jp,λ g(z)) = g(z).

Lemma 1. [7] Let n ∈ N∗ , α, β ∈ R, γ ∈ C with Re γ − αβ > 0. If P ∈ H[P (0), n]
with P (0) ∈ R and P (0) > α, then we have



zP ′ (z)
> α ⇒ Re P (z) > α, z ∈ U.
Re P (z) +
γ − βP (z)
Definition 1. [3, pg. 46], [4, pg. 228] Let c ∈ C with Re c > 0 and n ∈ N∗ . We
consider
" r
#
n
2Re c
|c| 1 +
+ Im c .
Cn = Cn (c) =
Re c
n
If the univalent function R : U → C is given by R(z) =

2Cn z
, then we denote
1 − z2

by Rc,n the ”Open Door” function, defined as
!
z+b
(z + b)(1 + b̄z)
Rc,n (z) = R
= 2Cn
,
1 + b̄z
(1 + b̄z)2 − (z + b)2
where b = R−1 (c).
43

A. Totoi - Meromorphic functions defined by a multiplier transformation

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
zg ′ (z) zϕ′ (z)
α
+
+ δ ≺ Rδ−pα,p (z).
g(z)
ϕ(z)
Φ,ϕ
If G = Jp,α,β,γ,δ
(g) is defined by

G(z) =

Φ,ϕ
Jp,α,β,γ,δ
(g)(z)



γ − pβ
= γ
z Φ(z)

Z

z

α

g (t)ϕ(t)t

δ−1

0

1

β

dt

,

(1)

then G ∈ Σp with z p G(z) 6= 0, z ∈ U, and


zG′ (z) zΦ′ (z)
Re β
+
+ γ > 0, z ∈ U.
G(z)
Φ(z)
All powers in (1) are principal ones.
Taking β = α = 1, δ = γ, Φ = ϕ ≡ 1, in the above theorem, and using the
1,1
notation Jp,γ instead of Jp,1,1,γ,γ
, we obtain the corollary:
Corollary 1. Let p ∈ N∗ , γ ∈ C with Re γ > p and let the function g ∈ Σp satisfying
the condition
zg ′ (z)
+ γ ≺ Rγ−p,p (z), z ∈ U.
g(z)
Then
Z
γ−p z
G(z) = Jp,γ (g)(z) =
g(t)tγ−1 dt ∈ Σp ,
zγ
0


′ (z)
zG
> 0, z ∈ U.
with z p G(z) 6= 0, z ∈ U, and Re γ +
G(z)
2. Main results

Definition 2. For p ∈ N∗ , n ∈ Z, λ ∈ C, Re λ > p and α < p < δ we define




′ 


n g(z)


z
J
p,λ


n
,
>
α,
z
∈
U
ΣSp,λ (α) = g ∈ Σp : Re −

n g(z)


Jp,λ






′ 


n


 z Jp,λ g(z) 
n
ΣSp,λ (α, δ) = g ∈ Σp : α < Re −
.
<
δ,
z
∈
U

n g(z)


Jp,λ


44

A. Totoi - Meromorphic functions defined by a multiplier transformation

n (α) if and only if J n g ∈ Σ∗ (α), respectively
Remark 3.
1. We have g ∈ ΣSp,λ
p
p,λ
n
n g ∈ Σ∗ (α, δ).
g ∈ ΣSp,λ (α, δ) if and only if Jp,λ
p
n g(z))′ = (λ − p)J n−1 g(z) − λJ n g(z), we can easy see
2. Using the equality z(Jp,λ
p,λ
p,λ
that for Re λ > p the condition


′ 
n
 z Jp,λ g(z) 
α < Re −
 < δ, z ∈ U,
n g(z)
Jp,λ

is equivalent to

"

Re λ − δ < Re (λ − p)

n−1
Jp,λ
g(z)
n g(z)
Jp,λ

#

< Re λ − α, z ∈ U.

(2)

3. We have
0
ΣSp,λ
(α, δ) = Σ∗p (α, δ),


Z
λ − p z λ−1
∗
1
t g(t)dt ∈ Σp (α, δ) .
ΣSp,λ (α, δ) = g ∈ Σp : G(z) =
zλ
0
n (α) and
The following theorem gives us a connection between the sets ΣSp,λ
n−1
n (α, δ) and ΣS n−1 (α, δ).
ΣSp,λ
(α), respectively between ΣSp,λ
p,λ

Theorem 2. Let p ∈ N∗ , n ∈ Z, λ ∈ C with Re λ > p, α < p < δ and g ∈ Σp . Then
n−1
n
g ∈ ΣSp,λ
(α) ⇔ Jp,λ (g) ∈ ΣSp,λ
(α),

respectively
n−1
n
g ∈ ΣSp,λ
(α, δ) ⇔ Jp,λ (g) ∈ ΣSp,λ
(α, δ),
Z z
λ−p
where Jp,λ (g)(z) =
tλ−1 g(t)dt.
zλ
0

n (α, δ) is equivalent to J n g ∈ Σ∗ (α, δ). Since
Proof . We know that g ∈ ΣSp,λ
p
p,λ
n−1 1
n−1 1
n
∗
Jp,λ g = Jp,λ (Jp,λ g), we have Jp,λ (Jp,λ g) ∈ Σp (α, δ), which is equivalent to
n−1
1
Jp,λ
(g) ∈ ΣSp,λ
(α, δ).

45

A. Totoi - Meromorphic functions defined by a multiplier transformation
n−1
1 (g) = J
Since Jp,λ
p,λ (g), we obtain Jp,λ (g) ∈ ΣSp,λ (α, δ). Therefore,
n−1
n
g ∈ ΣSp,λ
(α, δ) ⇔ Jp,λ (g) ∈ ΣSp,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] Let p ∈ N∗ , γ ∈ C and α < p < δ ≤ Re γ. If g ∈ Σ∗p (α, δ), then
G = Jp,γ (g) ∈ Σ∗p (α, δ).

Theorem 3. Let p ∈ N∗ , n ∈ Z, λ, γ ∈ C with Re λ > p and α < p < δ ≤ Re γ.
Then
n
n
g ∈ ΣSp,λ
(α, δ) ⇒ Jp,γ (g) ∈ ΣSp,λ
(α, δ).

n (α, δ) we have J n (g) ∈ Σ∗ (α, δ), hence, from Corollary 2,
Proof . Because g ∈ ΣSp,λ
p
p,λ
we obtain
n
Jp,γ (Jp,λ
(g)) ∈ Σ∗p (α, δ).
1 (J n (g)) = J n (J 1 (g)), where J 1 (g) = J (g), we obtain
Using the fact that Jp,γ
p,γ
p,γ
p,λ
p,λ p,γ
n
Jp,λ
(Jp,γ (g)) ∈ Σ∗p (α, δ),
n (α, δ).
which is equivalent to Jp,γ (g) ∈ ΣSp,λ

Corollary 3. Let n ∈ Z, p ∈ N∗ , λ ∈ C and α < p < δ ≤ Re λ. Then we have
n+1
n
ΣSp,λ
(α, δ) ⊂ ΣSp,λ
(α, δ).

n (α, δ). Taking γ = λ in Theorem 3 we have J
n
Proof . Let g ∈ ΣSp,λ
p,λ (g) ∈ ΣSp,λ (α, δ),
n+1
which, from Theorem 2, is equivalent to g ∈ ΣSp,λ
(α, δ). And the result follows.

46

A. Totoi - Meromorphic functions defined by a multiplier transformation

A theorem presented in [7] holds that:
Theorem 4. [7] Let p ∈ N∗ , β > 0, γ ∈ C and α < p <

Re γ
≤ δ.
β

If g ∈ Σ∗p (α, δ), with
β

zg ′ (z)
+ γ ≺ Rγ−pβ,p (z), z ∈ U,
g(z)

then G = Jp,β,γ (g) ∈ Σ∗p (α, δ).
Using this theorem, for β = 1, we get the next result.
Theorem 5. Let n ∈ Z, p ∈ N∗ , λ, γ ∈ C with Re λ > p and α < p < Re γ ≤ δ. If
n
g ∈ ΣSp,λ
(α, δ) and satisfies the condition
h
i′
n (g)(z)
z Jp,λ
n (g)(z)
Jp,λ

+ γ ≺ Rγ−p,p (z), z ∈ U,

n (α, δ).
then Jp,γ (g) ∈ ΣSp,λ

We omit the proof because it is similar with that of Theorem 3.
If we consider in Theorem 5 that δ → ∞ we get:
Theorem 6. Let n ∈ Z, p ∈ N∗ , λ, γ ∈ C with Re λ > p and α < p < Re γ. If
n
g ∈ ΣSp,λ
(α) and satisfies the condition
h
i′
n (g)(z)
z Jp,λ
n (g)(z)
Jp,λ

+ γ ≺ Rγ−p,p (z), z ∈ U,

n (α).
then Jp,γ (g) ∈ ΣSp,λ

Theorem 7. Let n ∈ N, p ∈ N∗ , λ, γ ∈ C and α < p < Re γ ≤ Re λ ≤ δ. If
n
h ∈ ΣSp,λ
(α, δ) and satisfies the condition
zh′ (z)
+ γ ≺ Rγ−p,p (z), z ∈ U,
h(z)
n (α, δ).
then Jp,γ (h) ∈ ΣSp,λ

47

A. Totoi - Meromorphic functions defined by a multiplier transformation

Proof . Let us denote H = Jp,γ (h). Because the conditions of Corollary 1 are fullfilled, we have H ∈ Σp with z p H(z) 6= 0, z ∈ U, and


zH ′ (z)
Re
+ γ > 0, z ∈ U.
(3)
H(z)


zH ′ (z)
+ λ > 0, z ∈ U.
Since Re γ ≤ Re λ, we obtain from (3) that Re
H(z)
Using now Corollary 1, we have Jp,λ H ∈ Σp with z p Jp,λ H(z) 6= 0, z ∈ U , and


z(Jp,λ H)′ (z)
+ λ > 0, z ∈ U.
Re
Jp,λ H(z)
n H(z) 6= 0, z ∈ U , and
By induction, we obtain z p Jp,λ

Re

"

n H)′ (z)
z(Jp,λ
n H(z)
Jp,λ

#

+ λ > 0, z ∈ U.

Since Re λ ≤ δ , we get from (4) that
"
#
n H)′ (z)
z(Jp,λ
Re −
< δ, z ∈ U.
n H(z)
Jp,λ

(4)

(5)

From the definition of H we have
γH(z) + zH ′ (z) = (γ − p)h(z), z ∈ U̇ .

(6)

n to (6) and after using the properties of J n , we obtain
We apply the operator Jp,λ
p,λ
n
n
n
(γ − p)Jp,λ
h(z) = γJp,λ
H(z) + z(Jp,λ
H(z))′ .

(7)

n
From h ∈ ΣSp,λ
(α, δ), we have

"

α < Re −

n h(z))′
z(Jp,λ
n h(z)
Jp,λ

#

< δ,

which is equivalent to (see Remark 3, item 2)
"
#
n−1
Jp,λ
h(z)
Re λ − δ < Re (λ − p) n
< Re λ − α.
Jp,λ h(z)

48

(8)

A. Totoi - Meromorphic functions defined by a multiplier transformation

From (7), we have
(λ − p)

n−1
Jp,λ
h(z)
n h(z)
Jp,λ

Let us denote P (z) = −

= (λ − p)

n H(z))′
z(Jp,λ
n H(z)
Jp,λ

n−1
n−1
γJp,λ
H(z) + z(Jp,λ
H(z))′
n H(z) + z(J n H(z))′
γJp,λ
p,λ

(9)

.

. Using the fact that

n−1
n
n
H(z),
z(Jp,λ
H(z))′ = (λ − p)Jp,λ
H(z) − λJp,λ

we obtain
P (z) = (p − λ)
Hence

n−1
Jp,λ
H(z)
n H(z)
Jp,λ

n−1
Jp,λ
H(z)
n H(z)
Jp,λ

=

+ λ, z ∈ U.

P (z) − λ
.
p−λ

(10)

If we apply the logarithmic differential to (10), we get
n−1
z(Jp,λ
H(z))′
n−1
Jp,λ
H(z)

so,

−

n−1
z(Jp,λ
H(z))′
n−1
Jp,λ
H(z)

n H(z))′
z(Jp,λ

=

n H(z)
Jp,λ

= −P (z) +

zP ′ (z)
,
P (z) − λ

zP ′ (z)
.
P (z) − λ

(11)

Using (10) and (11) we obtain from (9),

(λ − p)

= (λ − p)

n−1
h(z)
Jp,λ
n h(z)
Jp,λ

P (z) − λ
·
p−λ

= (λ − p)

n−1
H(z)
Jp,λ
n H(z)
Jp,λ

γ+
·
γ+

n−1
z(Jp,λ
H(z))′
n−1
Jp,λ
H(z)
n H(z))′
z(Jp,λ

=

n H(z)
Jp,λ

zP ′ (z)
zP ′ (z)
P (z) − λ
= λ − P (z) +
.
γ − P (z)
P (z) − γ

γ − P (z) +

From (8) and (12), we get


zP ′ (z)
< Re λ − α,
Re λ − δ < Re λ − P (z) +
P (z) − γ
49

(12)

A. Totoi - Meromorphic functions defined by a multiplier transformation

which is equivalent to


zP ′ (z)
< δ.
α < Re P (z) +
γ − P (z)

(13)

Because we know from (5) that Re P (z) < δ, z ∈ U, we have only to verify that
Re P (z) > α, z ∈ U. To show this we will use Lemma 1.
n H ∈ Σ and since we have proved that z p J n H(z) 6= 0, z ∈ U ,
We know that Jp,λ
p
p,λ
we get that P is analytic in U . We also have Re γ − α > 0 and P (0) = p > α. Since
the hypothesis of Lemma 1 is fulfilled for β = 1, we obtain Re P (z) > α, z ∈ U ,
which is equivalent to
"
#
n H(z))′
z(Jp,λ
Re −
> α, z ∈ U.
(14)
n H(z)
Jp,λ
n (α, δ).
Since H ∈ Σp , we get from (5) and (14) that H ∈ ΣSp,λ

If we consider γ = λ, in the above theorem, we obtain:
n (α, δ) with α < p < Re λ ≤ δ.
Corollary 4. Let n ∈ N, p ∈ N∗ , λ ∈ C and h ∈ ΣSp,λ
If
zh′ (z)
+ λ ≺ Rλ−p,p (z), z ∈ U.
h(z)
n (α, δ).
Then Jp,λ (h) ∈ ΣSp,λ

Taking n = 0 in Corollary 4, we get:
Corollary 5. Let p ∈ N∗ , λ ∈ C and α < p < Re λ ≤ δ. If h ∈ Σ∗p (α, δ) with
zh′ (z)
+ λ ≺ Rλ−p,p (z), z ∈ U,
h(z)
then Jp,λ (h) ∈ Σ∗p (α, δ).
If in Corollary 5 we consider δ 7→ ∞ we have the next result:

50

A. Totoi - Meromorphic functions defined by a multiplier transformation

Corollary 6. Let p ∈ N∗ , λ ∈ C and α < p < Re λ. If h ∈ Σ∗p (α) with
zh′ (z)
+ λ ≺ Rλ−p,p (z), z ∈ U,
h(z)
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 Re λ > p and α < p < Re γ. If
n (α) with J (J n (g)(z)) 6= 0, z ∈ U̇ , then
g ∈ ΣSp,λ
p,γ p,λ
n
Jp,γ (g) ∈ ΣSp,λ
(α).

Proof . Let G = Jp,γ (g). We know from Remark 1 that if g ∈ Σp and Re γ > p,
then G ∈ Σp .
Let
n G(z))′
z(Jp,λ
P (z) = −
n G(z) , z ∈ U.
Jp,λ
n (G) = J n (J (g)) = J (J n (g)) 6= 0 ı̂n U̇ . So P ∈ H(U ).
We have Jp,λ
p,γ p,λ
p,λ p,γ
Making some calculations, similar to those from the proof of Theorem 7, we get:

−

n g(z))′
z(Jp,λ
n g(z)
Jp,λ

= P (z) +

zP ′ (z)
, z ∈ U.
γ − P (z)

n (α) we have
From g ∈ ΣSp,λ

"

Re −

n g(z))′
z(Jp,λ
n g(z)
Jp,λ

#

> α, z ∈ U,

which is equivalent to


zP ′ (z)
Re P (z) +
> α, z ∈ U.
γ − P (z)
Since P ∈ H(U ) with P (0) = p > α and Re γ > α, we have from Lemma 1 (when
β = 1) that Re P (z) > α, z ∈ U, hence
n
G = Jp,γ (g) ∈ ΣSp,λ
(α).

51

A. Totoi - Meromorphic functions defined by a multiplier transformation

Taking n = 0 and λ = 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 < Re γ.
If g ∈ Σ∗p (α) with z p Jp,γ (g)(z) 6= 0, z ∈ U , then Jp,γ (g) ∈ Σ∗p (α).

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[8] B.A. Uralegaddi, C. Somanatha, Certain differential operators for meromorphic functions, Houston J. Math., 17 (1991), 279-284.
Alina Totoi
Department of Mathematics
”Lucian Blaga” University of Sibiu
Str. I. Raţiu, no. 5-7, Sibiu, România
email:totoialina@yahoo.com

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