Paper-01-22-2010. 112KB Jun 04 2011 12:03:08 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 22/2010
pp. 7-15

CLASSES OF MEROMORPHIC FUNCTIONS WITH RESPECT TO
N-SYMMETRIC POINTS

Maslina Darus, Rabha W. Ibrahim, Nikola Tuneski and Acu Mugur
Abstract. New subclasses of meromorphic functions with respect to N-symmetric
points are defined and studied. Some properties of these classes are discussed. Moreover, subordination for meromorphic functions with respect to N-symmetric points
is established.
2000 Mathematics Subject Classification: 30C45.

1. Introduction
Let H be the class of functions analytic in U := {z ∈ C : |z| < 1} and H[a, n] be
the subclass of H consisting of functions of the form f (z) = a+an z n +an+1 z n+1 +... .
Let A be the subclass of H consisting of functions of the form f (z) = z + a2 z 2 + ...
and S = {f ∈ A : f is univalent ∈ U }. Let Σ be the class of meromorphic function
in U := {z ∈ C : 0 < |z| < 1}, which takes the form

∞

f (z) =

1 X
+
an z n , (z ∈ U ).
z
n=0

Also let Θ be the class of the analytic functions in U := {z ∈ C∞ : |z| > 1}, which
takes the form
∞
X
an
g(z) = z + a0 +
, (z ∈ U )
zn
n=1

such that g(∞) = ∞ and g ′ (∞) = 1.

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M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...
∗

Definition 1.1. A function f (z) ∈ Σ is belongs to the class S if and only if
ℜ{−

zf ′ (z)
} > 0, (z ∈ U ).
f (z)

And a function g ∈ Θ is belongs to the class S ∗ if and only if
ℜ{

zg ′ (z)
} > 0, (z ∈ U ).
g(z)

Note that some classes in Θ defined and established in [1,4].
Sakaguchi (see [6]) introduced the class of functions that are starlike with respect
to N -symmetric points, N = 1, 2, 3, . . . , as follows


zf ′ (z)
SSP N = f ∈ A : ℜ{
} > 0, z ∈ U ,
fN (z)
where
fN (z) = z +

∞
X

am·N +1 z m·N +1 .

m=2

In order to give geometric characterization of the class SSP N for N ≥ 2 we define
ε := exp(2πi/N ) and we consider the weighted mean of f ∈ A,
1
Mf,N (z) = PN −1
j=1

ε−j

·

N
−1
X

ε−j · f (εj z).

j=1

It is easy to verify that
N −1

f (z) − Mf,N (z)
1 X −j
=
·
ε · f (εj z) = fN (z)
N
N
j=0

and further
fN (εj z) = εj fN (z),
′
fN
(εj z)

=

′
fN
(z)

N −1
1 X ′ j
f (ε z),
=
N
j=0

′′
′′ j
(z) =
(ε z) = fN
εj fN

8

1
N

N

−1
X
j=0

εj f ′′ (εj z).

M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

Now, the class SSP N is a collection of functions f ∈ A such that for any r close to
1, r < 1, the angular velocity of f (z) about the point Mf,N (z0 ) is positive at z = z0
as z traverses the circle |z| = r in the positive direction. The authors received
results concerning inclusion properties, some sufficient conditions for starlikeness
with respect to N -symmetric points and sharp upper bound of the Fekete-Szeg¨o
functional |a3 − µa22 | over the classes SSP N and KN [7] where the class KN of
convex functions with respect to N -symmetric points is defined by
ℜ{

[zf ′ (z)]′
′ (z) } > 0,
fN

z ∈ U.

For N = 1 we receive the well-known class of starlike functions, S ∗ ≡ SSP 1 , such
that f (U ) is a starlike region with respect to the origin. And for N = 2 we receive
2f2 (z) = f (z) − f (−z), Mf,2 (z) = f (−z) and


zf ′ (z)
} > 0, z ∈ U
SSP 2 = f ∈ A : ℜ{
f (z) − f (−z)
is the class of starlike functions with respect to symmetric points.
Now we introduce classes of functions with N-symmetric points belong to the
classes Σ and Θ as follows
∗

S N (α) := {f ∈ Σ : ℜ{−

zf ′ (z)

} > α, z ∈ U , α < 1}
fN (z)

where N = 1, 2, 3, .. and
∞

1 X
an.N +1 z n.N +1 , (z ∈ U ).
fN (z) = +
z

(1)

n=0

And the class
S ∗N (α) := {g ∈ Θ : ℜ{

zg ′ (z)
} > α, z ∈ U , 0 ≤ α < 1}

gN (z)

where N = 1, 2, 3, .. and
gN (z) = z + a0 +

∞
X
an.N +1

n=1

z n.N +1

, (z ∈ U ).

(2)

Note that when N = 1, the class S ∗N (α) reduces to the class which investigated by
Mocanu et [4] and Acu [1].
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M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

Moreover, we define the following subclasses
C N (α) := {f ∈ Σ : ℜ{−
And
C N (α) := {g ∈ Θ : ℜ{

(zf ′ (z))′
′ (z) } > α, z ∈ U , α < 1}.
fN

(zg ′ (z))′
′ (z) } > α, z ∈ U , 0 ≤ α < 1}.
gN

2. Some properties
∗

In this section, we study some properties of the classes S N , S ∗N , C N , and C N in
the following results.
Theorem 2.1. Let f ∈ Σ. Then
∗

∗

f (z) ∈ S N (α) implies fN (z) ∈ S (α),
∗

∗

where S (α) := S 1 (α) is the class of starlike functions in U .
∗

Proof. Suppose that f (z) ∈ S N (α), then from the definition we have
ℜ{−

zf ′ (z)
} > α, (z ∈ U ).
fN (z)

Substituting z by zεj , where εj = 1, j = 0, 1, ..., N − 1 yields
zf ′ (z)
} > α, (z ∈ U ) ⇒
fN (z)
zεj f ′ (zεj )
ℜ{−
} > α, (z ∈ U ) ⇒
fN (zεj )
zεj f ′ (zεj )
ℜ{−
} > α, (z ∈ U ) ⇒
fN (z)
P −1 j ′ j
z N
j=0 ε f (zε )
ℜ{−
} > α, (z ∈ U ) ⇒
fN (z)
zf ′ (z)
} > α, (z ∈ U ).
ℜ{− N
fN (z)
ℜ{−

∗

Hence fN (z) ∈ S (α).
By using the same method of Theorem 2.1, we have the following result.

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M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

Theorem 2.2. Let g ∈ Θ. Then
g(z) ∈ S ∗N (α) implies gN (z) ∈ S ∗ (α),
where S ∗ (α) := S ∗1 (α) is the class of starlike functions in U .
Theorem 2.3. Let f ∈ Σ. Then
f (z) ∈ C N (α) implies fN (z) ∈ C(α),
where C(α) := C 1 (α) is the class of convex functions in U .
Proof. Suppose that f (z) ∈ C N (α), then from the definition we have
ℜ{−

(zf ′ (z))′
′ (z) } > α, (z ∈ U ).
fN

Substituting z by zεj , where εj = 1, j = 0, 1, ..., N − 1 yields

′

′′
fN (zεj ) + zεj fN (zεj )
} > α, (z ∈ U ) ⇒
ℜ{−
(fN (zεj ))′

′

′′
fN (zεj ) + zεj fN (zεj )
ℜ{−
} > α, (z ∈ U ) ⇒
(fN (z))′




PN −1
PN −1 j
j ′′
j ′
j=0 fN (zε ) + z
j=0 ε fN (zε )
ℜ{−
} > α, (z ∈ U ) ⇒
(fN (zεj ))′
z(fN (z))′′ + (fN (z))′
ℜ{−
} > α, (z ∈ U ) ⇒
(fN (z))′
(zf ′ (z))′
} > α, (z ∈ U )
ℜ{− ′N
fN (z)
Hence fN (z) ∈ C(α).
By using the same method of Theorems 2.3, we have the following result.
Theorem 2.4. Let g ∈ Θ. Then
g(z) ∈ C N (α) implies gN (z) ∈ C(α).
where C(α) := C 1 (α) is the class of convex functions in U .
Theorem 2.5. Let f ∈ Σ. Then
∗

f (z) ∈ C N (α) ⇔ zf ′ (z) ∈ S N (α).
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M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

′ (z) and
Proof. Let h(z) = zf ′ (z). Then hN (z) = zfN

f ∈ C N (α) ⇔ ℜ{−

zh′ (z)
[zf ′ (z)]′
∗
} > α ⇔ zf ′ (z) ∈ S N (α).
}
>
α
⇔
ℜ{−
′ (z)
fN
hN (z)

In the same manner, we can receive the following result
Theorem 2.6. Let g ∈ Θ. Then
g(z) ∈ C N (α) ⇔ zg ′ (z) ∈ S ∗N (α).

3. Subordination results
Recall that the function F is subordinate to G, written F ≺ G, if G is univalent,
F (0) = G(0) and F (U ) ⊂ G(U ). In general, given two functions F (z) and G(z),
which are analytic in U, the function F (z) is said to be subordination to G(z)
in U if there exists a function h(z), analytic in U with h(0) = 0 and |h(z)| <
1 f or all z ∈ U such that F (z) = G(h(z)) f or all z ∈ U (see[2,3]).
We need to the following result in sequel.
Lemma 3.1. [2] Let q(z) be univalent in the unit disk U and θ and φ be
analytic in a domain D containing q(U ) with φ(w) 6= 0 when w ∈ q(U ). Set
Q(z) := zq ′ (z)φ(q(z)), h(z) := θ(q(z)) + Q(z). Suppose that
1. Q(z) is starlike univalent in U , and
′ (z)
2. ℜ zh
Q(z) > 0 for z ∈ U.
If p(z) is analytic in U and
θ(p(z)) + zp′ (z)φ(p(z)) ≺ θ(q(z)) + zq ′ (z)φ(q(z))
then p(z) ≺ q(z) and q(z) is the best dominant.
Theorem 3.1. Let q(z) be univalent and q(z) 6= 0 in U and
′ (z)
is starlike univalent in U, and
(1) zqq(z)
(2) ℜ{1 +
If f ∈ Σ and

zq ′′ (z)
q ′ (z)

−[(1 − γ)

−

zq ′ (z)
q(z)

−

q(z)
γ }

> 0 for z ∈ U, γ 6= 0.

zf ′′ (z)

zq ′ (z)
zf ′ (z)
+γ 1+ ′
,
] ≺ q(z) − γ
fN (z)
fN (z)
q(z)

then
−

zf ′ (z)
≺ q(z)
fN (z)
12

(3)

M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

and q(z) is the best dominant.
Proof. Define the function p(z) by
p(z) := −

zf ′ (z)
.
fN (z)

A computation gives
p(z) − γ

zf ′ (z)
zf ′′ (z)

zp′ (z)
]
= −[(1 − γ)
+γ 1+ ′
p(z)
fN (z)
fN (z)

where the functions θ and φ defined by
γ
θ(ω) := ω and φ(ω) := − .
ω
It can be easily observed that θ(ω) is analytic in C and φ(ω) is analytic in C\{0}
and that φ(ω) 6= 0 when ω ∈ C\{0}. Also, by letting
Q(z) = zq ′ (z)φ(q(z)) = −γz

q ′ (z)
q(z)

and
h(z) = θ(q(z)) + Q(z) = q(z) − γz

q ′ (z)
,
q(z)

we find that Q(z) is starlike univalent in U and that
ℜ{

zh′ (z)
zq ′′ (z) zq ′ (z) q(z)
} = ℜ{1 + ′
−
−
} > 0.
Q(z)
q (z)
q(z)
γ

Then the relation (5) follows by an application of Lemma 3.1.
The next result can be found in [5].
Corollary 3.1. Let the assumptions of Theorem 3.1 hold. Then
−

zf ′ (z)
≺ q(z)
f (z)

(4)

and q(z) is the best dominant.
Proof. By letting N = 1 in the above theorem the result follows immediately.
In Theorem 3.1, let
q(z) :=

1 + (1 − 2α)z
,
1−z
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M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

we have the following result
Corollary 3.2. Let α < 0, γ 6= 0. If f ∈ Σ and
−[(1 − γ)

zf ′′ (z)

1 + 2[1 − γ + (α − 1)γ]z + (1 − 2α)2 z 2
zf ′ (z)
+γ 1+ ′
,
]≺
fN (z)
fN (z)
1 − 2αz − (1 − 2α)z 2

then
−ℜ{

zf ′ (z)
} > α.
fN (z)

(5)

Note that when N = 1, Corollary 3.2. reduces to result in [5, Corollary 3.1].
Acknowledgement: The work of the first two authors was supported by eScienceFund:
04-01-02-SF0425, MOSTI, Malaysia. The work of the third author was supported by The
Ministry of Education and Science of the Republic of Macedonia (Research Project No.171383/1). Also, this paper is part of the activities contained in the Romanian-Malaysian
Mathematical Research Center.

References
[1] Acu M., Some subclasses of meromorphic functions, Filomat 21:2(2007), 1-9.
[2] Miller S.S., Mocanu P. T., Differential Subordinations. Theory and Applications, Marcel Dekker, New York-Basel, 2000.
[3] Miller S.S., Mocanu P.T., Subordinants of differential superordinations, Complex Variables, 48(10)(2003), 815-826.
[4] Mocanu P. T., Bulboaca T., Sˇ
alˇagean G.S., Teoria geometrica a functiilor
univalente, Casa Cartii de Stiinta(Cluj), 1999.
[5] Ravichandran V., Kumar S., Darus M., On a subordination theorem for a
class of meromorphic functions, J. Ineq. Pure and Appli. Math., 5(8)(2004),1-4.
[6] Sakaguchi, K., On a certain univalent mapping, J. Math. Soc. Japan 11
(1959), 72-75.
[7] Tuneski N., Darus M., Ibrahim R.W., Some results for univalent functions
defined with respect to N -symmetric points, submited.
Maslina Darus, Rabha W. Ibrahim
School of Mathematical Sciences, Faculty of science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600, Selangor Darul Ehsan, Malaysia
email: maslina@ukm.my , rabhaibrahim@yahoo.com
Nikola Tuneski
Faculty of Mechanical Engineering

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M. Darus, R. Ibrahim, N. Tuneski, M. Acu - Classes of meromorphic functions ...

Karpoˇs II b.b., 1000 Skopje, Republic of Macedonia
email: nikolat@mf.edu.mk
Acu Mugur Alexandru
Department of Mathematics
University ”Lucian Blaga” of Sibiu
Str. Dr. I. Rat.iu, No. 5-7, 550012 - Sibiu, Romania
email: acu mugur@yahoo.com

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