06-21.2010. 113KB Jun 04 2011 12:07:14 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 21/2010
pp. 55-63

A NEW CLASS OF HARMONIC MULTIVALENT FUNCTIONS
DEFINED BY AN INTEGRAL OPERATOR

˘
Luminit
¸ a-Ioana Cotˆırla
Abstract. We define and investigate a new class of harmonic multivalent
functions defined by S˘al˘
agean integral operator. We obtain coefficient inequalities
and distortion bounds for the functions in this class.
2000 Mathematics Subject Classification: 30C45, 30C50, 31A05.
Keywords and phrases: Harmonic univalent functions, integral operator, distortion inequalities.
1. Introduction
For fixed positive integer p, denote by H(p) the set of all harmonic multivalent
functions f = h+g which are sense-preserving in the open unit disk U = {z : |z| < 1}

where h and g are of the form
p

h(z) = z +

∞
X

ak+p−1 z

k+p−1

,

g(z) =

k=2

∞
X

bk+p−1 z k+p−1 ,

|bp | < 1.

(1)

k=1

The integral operator I n was introduced by S˘
al˘
agean [9]. For fixed positive
integer n and for f = h + g given by (1) we define the modified S˘al˘
agean operator
n
I f as
(2)
I n f (z) = I n h(z) + (−1)n I n g(z); p > n, z ∈ U
where
I n h(z) = z p +

∞
X
k=2

and
I n g(z) =

∞
X
k=1


n
p
ak+p−1 z k+p−1
k+p−1


n
p
bk+p−1 z k+p−1 .

k+p−1
55

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...

It is known that, (see[3]), the harmonic function f = h + g is sense- preserving
in U if |g ′ | < |h′ | in U. The class H(p) was studied by Ahuja and Jahangiri [1] and
the class H(p) for p = 1 was defined and studied by Jahangiri et al. in [6].
For fixed positive integers n, p, and for 0 ≤ α < 1, β ≥ 0 we let Hp (n+1, n, α, β)
denote the class of multivalent harmonic functions of the form (1) that satisfy the
condition

 I n f (z)
 I n f (z)
 + α.

Re
−
1
(3)

>
β
I n+1 f (z)
I n+1 f (z)
The subclass Hp− (n + 1, n, α, β) consists of functions fn = h + gn in Hp (n, α, β)
so that h and g are of the form
p

h(z) = z −

∞
X

ak+p−1 z

k+p−1

n−1

gn (z) = (−1)

,

k=2

∞
X

bk+p−1 z k+p−1 ,

|bp | < 1. (4)

k=1

The families Hp (n + 1, n, α, β) and Hp− (n + 1, n, α, β) include a variety of wellknown classes of harmonic functions as well as many new ones. For example
H1− (1, 0, α, 0) ≡ HS(α) is the class of sense-preserving, harmonic univalent functions f which are starlike of order α in U, H1− (2, 1, α, 0) ≡ HK(α) is the class of
sense-preserving, harmonic univalent functions f which are convex of order α in U
and H1− (n + 1, n, α, 0) ≡ H − (n, α) is the class os S˘al˘
agean type harmonic univalent
functions.

For the harmonic functions f of the form (1) with b1 = 0, Avei and Zlotkiewicz
[2] showed that if
∞
X
k(|ak | + bk |) ≤ 1,
k=2

then f ∈ HS(0) and if

∞
X

k 2 (|ak | + |bk |) ≤ 1,

k=2

then f ∈ HK(0). Silverman [10] proved that the above two coefficient conditions are
also necessary if f = h + g has negative coefficients. Later, Silverman and Silvia[11]
improved the results of [6] and [9] to the case b1 not necessarily zero.
For the harmonic functions f of the form (4) with n = 1, Jahangiri [5] showed

that f ∈ HS(α) if and only if
∞
X
k=2

(k − α)|ak | +

∞
X

(k + α)|bk | ≤ 1 − α

k=1

56

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...
and f ∈ H1− (2, 1, α, 0) if and only if
∞
X

k(k − α)|ak | +

k=2

∞
X

k(k + α)|bk | ≤ 1 − α.

k=1

In this paper, the coefficient conditions for the classes HS(α) and HK(α) are
extended to the class Hp (n + 1, n, α, β), of the form (3) above. Furthermore, we
determine distortion theorem for the functions in Hp− (n + 1, n, α, β).
2.Main results
In the first theorem, we introduce a sufficient coefficient bound for harmonic
functions in Hp (n + 1, n, α, β).
Theorem 2.1. Let f = h + g be given by (1). If
∞

X

{Ψ(n + 1, n, p, α, β)|ak+p−1 | + Θ(n + 1, n, p, α, β)|bk+p−1 |} ≤ 2,

k=1

where
Ψ(n + 1, n, p, α, β) =
and
Θ(n + 1, n, p, α, β) =
ap = 1,

0 ≤ α < 1,

β ≥ 0,


n
p
k+p−1 (1

+ β) − (β + α)


n+1
p
k+p−1

1−α

n
p
k+p−1 (1

+ β) +


n+1
p
(β
k+p−1

+ α)

1−α
n ∈ N. Then f ∈ Hp (n + 1, n, p, α, β).

Proof. According to (2) and (3) we only need to show that
Re

I n f (z) − αI n+1 f (z) − βeiΘ |I n f (z) − I n+1 f (z)|

≥ 0.
I n+1 f (z)

The case r = 0 is obvious. For 0 < r < 1 it follows that
Re

I n f (z) − αI n+1 f (z) − βeiθ |I n f (z) − I n+1 f (z)|

=
I n+1 f (z)
(1−α)z p +

= Re


zp +

∞
X

∞
X



ak+p−1 z k+p−1 Γn − αΓn+1

k=2

Γn+1 ak+p−1 z k+p−1 + (−1)n+1

∞
X
k=1

k=2

57

Γn+1 bk+p−1 z k+p−1

,

,

(5)

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...
∞
X

(−1)n

+
zp +

βeiΘ |

∞
X



bk+p−1 z k+p−1 Γn + αΓn+1

k=1
n+1

Γ

ak+p−1 z

k+p−1

∞
X

−
n+1

Γ

bk+p−1 z

k+p−1

k=1

∞
X




bk+p−1 z k+p−1 Γn + Γn+1 |
ak+p−1 z k+p−1 Γn − Γn+1 + (−1)n

zp +

∞
X

k=1
∞
X
n+1

Γn+1 ak+p−1 z k+p−1 + (−1)

k=2
1−α+

= Re



∞
X

1+

n+1

Γ

(−1)n
∞
X

∞
X

ak+p−1 z

∞
X

k=1

k−1

Γ

∞
X

n+1

+ (−1)

Γn+1 bk+p−1 z k+p−1 z −p

k=1



bk+p−1 z k−1+p z −p Γn + αΓn+1

k=1
n+1

Γn+1 bk+p−1 z k+p−1





ak+p−1 z k−1 Γn − αΓn+1

k=2

k=2

1+

+ (−1)

k=2
∞
X
k=2

+

n+1

ak+p−1 z

k−1

n+1

+ (−1)

k=2

∞
X

−
n+1

Γ

bk+p−1 z

k+p−1 −p

z

k=1

∞
∞
X
X

 n

 n

n+1
k+p−1
n
iΘ
−p

Γ + Γn+1 bk+p−1 z k+p−1 
Γ −Γ
ak+p−1 z
+ (−1)
βe z
k=2
1+

∞
X

Γn+1 ak+p−1 z k−1 + (−1)n+1

k=2

∞
X

k=1

Γn+1 bk+p−1 z k+p−1 z −p



k=1

(1 − α) + A(z)
= Re
,
1 + B(z)
where
Γ=

p
.
k+p−1

For z = reiΘ we have
iΘ

A(re ) =

∞
X

(An − αAn+1 )ak+p−1 rk−1 e(k−1)Θi +

k=2

+(−1)n

∞
X

(An + An+1 α)bk+p−1 rk−1 e−(k+2p−1)Θi − βe−(p−1)iΘ D(n + 1, n, p, α),

k=1

58

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...

where
∞
X
(An − An+1 )ak+p−1 rk−1 e−(k+p−1)iΘ
D(n + 1, n, p, α) = = 
k=2

+ (−1)n

∞
X
k=1


(An + An+1 )bk+p−1 rk−1 e−(k+p−1)iΘ ,

and
B(reiΘ ) =

∞
X

An+1 ak+p−1 rk−1 e(k−1)Θi

k=2

+ (−1)n+1

∞
X

An+1 bk+p−1 rk−1 e−(k+2p−1)Θi .

k=1

Setting
1 + w(z)
1 − α + A(z)
= (1 − α)
.
1 + B(z)
1 − w(z)
The proof will be complete if we can show that |w(z)| ≤ r < 1. This is the case
since, by the condition (5), we can write:

|w(z)| = 
∞
X

[(1 + β)(An − An+1 )|ak+p−1 | + (1 + β)(An + An+1 )|bk+p−1 |]rk−1

k=1
∞
X

4(1−α)−

<
n

{[A (1 + β) − ΛA

k=1
∞
X

<


A(z) − (1 − α)B(z)
≤
A(z) + (1 − α)B(z) + 2(1 − z)

n+1

n

]|ak+p−1 | + [A (1 + β) + ΛA

n+1

]|bk+p−1 |}r

k−1

(1 + β)(An − An+1 )|ak+p−1 | + (Am + An+1 )(1 + β)|bk+p−1 |

k=1
4(1−α)−{

∞
X

≤
n

[A (1 + β) − ΛA

n+1

n

]|ak+p−1 | + [A (1 + β) + ΛA

n+1

]|bk+p−1 |}

k=1

≤ 1,
where Λ = β + 2α − 1.
The harmonic univalent functions
f (z) = z p +

∞
X
k=2

∞

X
1
1
yk z k+p−1 , (6)
xk z k+p−1 +
Ψ(n + 1, n, p, α, β)
Θ(n + 1, n, p, α, β)
k=1

59

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...

where n ∈ N, 0 ≤ α < 1, β ≥ 0 and

∞
X

|xk | +

k=2

∞
X

|yk | = 1, show that the coefficient

k=1

bound given by (5) is sharp.
The functions of the form (6) are in Hp (n + 1, n, α, β) because
∞
X

[Ψ(n + 1, n, p, α, β)|ak+p−1 | + Θ(n + 1, n, p, α, β)|bk+p−1 |] =

k=1

=1+

∞
X

|xk | +

k=2

∞
X

|yk | = 2.

k=1

In the following theorem it is show that the condition (5) is also necessary for
the function fn = h + g n , where h and gn are of the form (4).
Theorem 2.2. Let fn = h + g n be given by (4). Then fn ∈ Hp− (n + 1, n, α, β)
if and only if
∞
X

[Ψ(n + 1, n, p, α, β)ak+p−1 + Θ(n + 1, n, p, α, β)bk+p−1 ] ≤ 2,

(7)

k=1

ap = 1, 0 ≤ α < 1, n ∈ N.
Proof. Since Hp− (n + 1, n, α, β) ⊂ Hp (n + 1, n, α, β), we only need to prove the
”only if” part of the theorem. For functions fn of the form (4), we note that the
condition

 I n f (z)
 I n f (z)
Re
− 1 + α
> β  n+1
n+1
I
f (z)
I
f (z)
is equivalent to
(1 −
Re

α)z p

−

∞
X

(Γn − αΓn+1 )ak+p−1 z k+p−1

k=2


zp −

∞
X

Γn+1 ak+p−1 z k+p−1 + (−1)2n

k=2

∞
X

+
Γn+1 bk+p−1 z k+p−1

k=1

(−1)2n−1

∞
X

(Γn + Γn+1 α)bk+p−1 z k+p−1

k=1

+
zp −

∞
X

Γn+1 ak+p−1 z k+p−1 + (−1)2n

k=2

∞
X
k=1

60

Γn+1 bk+p−1 z k+p−1

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...

−

∞
∞
X

 X
βeiΘ  −
(Γn − Γn+1 )ak+p−1 z k+p−1 + (−1)2n−1
(Γn + Γn+1 )bk+p−1 z k+p−1 
k=2

zp −

∞
X

k=1
∞
X
2n
n+1

Γn+1 ak+p−1 z k+p−1 + (−1)

Γ

k=2



bk+p−1 z k+p−1

k=1

(8)
≥ 0,
p
p+k−1 .

where Γ =
The above required condition (8) must hold for all values of z ∈ U. Upon choosing
the values of z on the positive real axis where 0 ≤ z = r < 1, we must have
(1 − α) −

∞
X

[Γn (1 + β) − (β + α)Γn+1 ]ak+p−1 rk−1

k=2

1−

∞
X

n+1

Γ

ak+p−1 r

k−1

+

∞
X

1−

∞
X

(9)
n+1

Γ

bk+p−1 r

k+p−1

k=1

k=2

−

∞
X

[Γn (1 + β) + Γn+1 (β + α)]bk+p−1 rk−1

k=1

+

Γ

n+1

ak+p−1 r

k−1

+

k=2

∞
X

≥ 0.
n+1

Γ

bk+p−1 r

k−1

k=1

If the condition (8) does not hold, then the expression in (9) is negative for r sufficiently close to 1. Hence there exist z0 = r0 in (0, 1) for which the quotient in (9) is
negative. This contradicts the required condition for fn ∈ Hp− (n + 1, n, α, β). And
so the proof is complete.
The following theorem gives the distortion bounds for functions in Hp− (n +
1, n, α, β) which yields a covering results for this class.
Theorem 2.3. Let fn ∈ Hp− (n + 1, n, α, β). Then for |z| = r < 1 we have
|fn (z)| ≤ (1 + bp )rp + [Φ(n + 1, n, p, α, β) − Ω(n + 1, n, p, α, β)bp ]rn+1+p
and
|fn (z)| ≥ (1 − bp )rp − {Φ(n + 1, n, p, α, β) − Ω(n + 1, n, p, α, β)bp }rn+p+1
where,
Φ(n + 1, n, p, α, β) =

1−α
p
n
p+1 (1

61

+ β) −

p
n+1
(β
p+1

+ α)

,

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...
(1 + β) + (α + β)
.
p
n+1
+ β) − p+1
(β + α)

Ω(n + 1, n, p, α, β) =

p
n
p+1 (1

Proof. We prove the right side inequality for |fn |. The proof for the left hand
inequality can be done using similar arguments. Let fn ∈ Hp− (n + 1, n, α, β). Taking
the absolute value of fn then by Theorem 2.2, we can obtain:
p

|fn (z)| = |z −

∞
X

ak+p−1 z

k+p−1

n−1

+ (−1)

k=2

≤ rp +

∞
X

∞
X

bk+p−1 z k+p−1 | ≤

k=1

ak+p−1 rk+p−1 +

k=2

∞
X

bk+p−1 rk+p−1 =

k=1

= r p + bp r p +

∞
X

(ak+p−1 + bk+p−1 )rk+p−1 ≤

k=2
∞
X

≤ r p + bp r p +

= (1 + bp )rp + Φ(n + 1, n, p, α, β)

(ak+p−1 + bk+p−1 )rp+1 =

k=2
∞
X
k=2

1
(ak+p−1 + bk+p−1 )rp+1 ≤
Φ(n + 1, n, p, α, β)

p

≤ (1 + bp )r + Φ(n + 1, n, p, α, β)rn+p+1 ×
∞
X
×[
Ψ(n + 1, n, p, α, β)ak+p−1 + Θ(n + 1, n, p, α, β)bk+p−1 ] ≤
k=2

≤ (1 + bp )rp + [Φ(n + 1, n, p, α, β) − Ω(n + 1, n, p, α, β)bp ]rn+1+p .
The following covering result follows from the left hand inequality in Theorem
2.3.
Corollary 2.4. Let fn ∈ Hp− (n + 1, n, α, β), then for |z| = r < 1 we have
{w : |w < 1 − bp − [Φ(n + 1, n, p, α, β) − Ω(n + 1, n, p, α, β)bp ] ⊂ fn (U)}.
For β = 0 we obtain the results given in [4].
For β = 0, p = 1 and using the differential S˘al˘
agean operator we obtain the
results given [7].
The beautiful results, for harmonic functions, was obtained by P. T. Mocanu in
[8].

62

L.I. Cotˆırl˘a - A new class of harmonic multivalent functions defined by...

References
[1]O.P. Ahuja, J.M. Jahangiri, Multivalent harmonic starlike functions, Ann.
Univ. Marie Curie-Sklodowska Sect. A, LV 1(2001), 1-13.
[2]Y. Avci, E. Zlotkiewicz, On harmonic univalent mappings, Ann. Univ. Marie
Crie-Sklodowska, Sect. A., 44(1991).
[3]J. Clunie, T. Scheil- Small, Harmonic univalent functions, Ann. Acad. Sci.
Fenn. Ser. A. I. Math., 9(1984), 3-25.
[4] L. I. Cotˆırl˘a, Harmonic univalent functions defined by an integral operator,
Acta Universitatis Apulensis, 17(2009), 95-105.
[5]J. M. Jahangiri, Harmonic functions starlike in the unit disc, J. Math. Anal.
Appl., 235(1999).
[6]J. M. Jahangiri, G. Murugusundaramoorthy and K. Vijaya, S˘
al˘
agean harmonic
univalent functions, South. J. Pure Appl. Math. , 2(2002), 77-82.
[7]A. R. S. Juma, L. I. Cotˆırl˘a, On harmonic univalent functions defined by
generalized S˘
al˘
agean derivatives, submitted for publications.
[8]P. T. Mocanu, Three-cornered hat harmonic functions, Complex Variables and
Elliptic Equation, 12(2009), 1079-1084.
[9]G.S. S˘al˘
agean, Subclass of univalent functions, Lecture Notes in Math. SpringerVerlag, 1013(1983), 362-372.
[10]H. Silverman, Harmonic univalent functions with negative coefficients, J.
Math. Anal. Appl. 220(1998), 283-289.
[11]H. Silverman, E. M. Silvia, Subclasses of harmonic univalent functions, New
Zealand J. Math. 28(1999), 275-284.
Luminita-Ioana Cotˆırl˘
a
Babe¸s-Bolyai University
Faculty of Mathematics and Computer Science
400084 Cluj-Napoca, Romania
E-mail: uluminita@math.ubbcluj.ro, luminita.cotirla@yahoo.com

63