General Mathematics Vol. 14, No. 4 (2006), 37–46
A New Class of Multivalent Harmonic
Functions 1
K. Al Shaqsi
and
M. Darus
In memoriam of Associate Professor Ph. D. Luciana Lupaş
Abstract
In this paper, we introduce a new class of multivalent harmonic
functions. We investigate various properties of functions belonging
to this class. Coefficients bounds, distortion bounds and extreme
points are given.
2000 Mathematics Subject Classification: 30C45.
Keywords: Multivalent functions, harmonic functions, derivative
operator.
1
Introduction
A continuous functions f = u + iv is a complex valued harmonic function
in a complex domain C if both u and v are real harmonic in C. In any
1
Received 24 September, 2006
Accepted for publication (in revised form) 26 October, 2006
37
38
K. Al Shaqsi and M. Darus
simply connected domain D ⊂ C we can write f = h + g, where h and g
are analytic in D. We call h the analytic part and g the co-analytic part
of f . A necessary and sufficient condition for f to be locally univalent and
sense-preserving in D is that |h′ (z)| > |g ′ (z)| in D. See Clunie and SheilSmall (see [2]).
Denote by H(p) the class of functions f = h + g that are harmonic multivalent and sense-preserving in the unit disk U = {z : |z| < 1}. For
f = h + g ∈ H(p) we may express the analytic functions h and g as
(1)
h(z) = z p +
∞
X
ak z k ,
g(z) =
∞
X
bk z k ,
|bp | < 1.
k=p
k=p+1
Also denote by T (p), the subclass of H(p) consisting of all functions f =
h + g where h and g are given by
(2)
h(z) = z p −
∞
X
|ak |z k ,
g(z) = −
k=p+1
∞
X
|bk |z k ,
|bp | < 1.
k=p
We denote by Hλn (p, α) the class of all functions of the form (1.1) that satisfy
the condition
(
(Dλn+p−1 f (z))′
ℜ
pz p−1
(3)
)
> α,
where 0 ≤ α < p, p ∈ N, λ ≥ 0, n ∈ N0 and Dλn+p−1 f (z) = Dλn+p−1 h(z) +
Dλn+p−1 g(z).
When p = 1, Dλn denotes the operator introduced by [3]. For h and g given
by (1.1) we have
Dλn+p−1 h(z)
∞
X
£
¤
=z +
1 + λ(k − p) C(n, k, p)ak z k ,
p
k=p+1
A New Class of Multivalent Harmonic Functions
Dλn+p−1 g(z)
39
∞
X
£
¤
=
1 + λ(k − p) C(n, k, p)bk z k
k=p
where λ ≥ 0, p ∈ N, n > −p and C(n, k, p) =
¡k+n−1¢
n+p−1
.
Note that :
∗
H00 (1, 0) ≡ SH
studied by Silverman [1],
Hλ0 (1, 0) ≡ H(λ) studied by Yalçin and Öztürk [7],
H00 (1, α) ≡ NH (α) studied by Ahuja and Jahangiri [5],
Hλn (1, 0) ≡ Hλn studied by the authors in [4].
Also we note that for the analytic part the class H0n (p, α) was introduced
and studied by Goel and Sohi [6].
We further denote by Tλn (p, α) the subclass of Hλn (p, α), where
Tλn (p, α) = T (p) ∩ Hλn (p, α).
2
Coefficients Bounds
Theorem 2.1. Let f = h + g with h and g are given by (1.1). Let
(4)
∞
X
£
¤
k 1 + λ(k − p) C(n, k, p)[|ak | + |bk |] ≤ p(2 − α)
k=p
where ap = p, λ ≥ 0 and 0 ≤ α < p. Then f is harmonic multivalent sense
preserving in U and f ∈ Hλn (p, α).
Proof. Letting w(z) =
n+p−1
(Dλ
f (z))′
p−1
pz
. Using the fact ℜ{w} ≥ α if and only
40
K. Al Shaqsi and M. Darus
if |p − α + w(z)| ≥ |p + α − w(z)|, it suffices to show that
¯
¯ ¯
¯
n+p−1
n+p−1
¯
¯
′¯
′¯
(D
f
(z))
(D
f
(z))
¯
¯ ¯
¯
λ
λ
(5)
¯p − α +
¯ − ¯p + α −
¯ ≥ 0.
¯
¯ ¯
¯
pz p−1
pz p−1
Substituting for h and g in (2.2) yields
¯
¯
n+p−1
¯
n+p−1
′
′¯
(D
g(z))
(D
h(z))
+
¯
¯
λ
λ
¯−
¯p − α +
¯
¯
pz p−1
¯
¯
n+p−1
n+p−1
¯
′
′
g(z)) ¯¯
(Dλ
h(z)) + (Dλ
¯
−¯p + α −
¯=
¯
¯
pz p−1
¯
∞
¯
X
¤
k£
¯
= ¯p + 1 − α +
1 + λ(k − p) C(n, k, p)ak z k−p +
¯
p
k=p+1
¯
∞
¯
X
£
¤
k
k−p ¯
+
1 + λ(k − p) C(n, k, p)bk z ¯−
¯
p
k=p
¯
∞
¯
X
¤
k£
¯
1 + λ(k − p) C(n, k, p)ak z k−p −
−¯p − 1 + α −
¯
p
k=p+1
¯
∞
¯
X
¤
k£
¯
1 + λ(k − p) C(n, k, p)bk z k−p ¯ ≥
−
¯
p
k=p
(
" ∞
X k£
¤
≥ 2 (1 − α) −
1 + λ(k − p) C(n, k, p)|ak ||z k−p |+
p
k=p+1
#)
∞
X
£
¤
k
1 + λ(k − p) C(n, k, p)|bk ||z k−p |
>
+
p
k=p
(
" ∞
X £
¤
> 2 p(1 − α) −
k 1 + λ(k − p) C(n, k, p)|ak |+
k=p+1
#)
∞
X
£
¤
k 1 + λ(k − p) C(n, k, p)|bk |
> 0.
+
k=p
A New Class of Multivalent Harmonic Functions
41
The Harmonic mappings
∞
X
xk
¤
zk +
k
1
+
λ(k
−
p)
C(n,
k,
p)
k=p+1
p
f (z) = z +
+
∞
X
k=p
where
∞
P
|xk | +
k=p+1
∞
P
£
yk
¤
zk
k 1 + λ(k − p) C(n, k, p)
£
|yk | = p(1 − α), show that the coefficient bound given
k=p
by (2.1) is sharp.
The functions of the form (2.3) are in Hλn (p, α) because
∞
X
k=p+1
³
´
£
¤
k 1 + λ(k − p) C(n, k, p) |ak | + |bk | =
=p+
∞
X
|xk | +
k=2
∞
X
|yk | = p(2 − α).
k=1
The restriction placed in Theorem 2.1 on the moduli of the coefficients of
f = h + g enables us to conclude for arbitrary rotation of the coefficients
of f that the resulting functions would still be harmonic multivalent and
f ∈ Hλn (p, α).
We next show that the condition (2.1) is also necessary for functions in
Tλn (p, α).
Theorem 2.2. Let f = h + g with h and g are given by (1.2). Then
f ∈ Tλn (p, α) if and only if
(6)
∞
X
£
¤
k 1 + λ(k − p) C(n, k, p)[|ak | + |bk |] ≤ p(2 − α)
k=p
where ap = p, λ ≥ 0 and 0 ≤ α < p.
42
K. Al Shaqsi and M. Darus
Proof. The ”if ” part follows from Theorem 2.1 upon noting Tλn (p, α) ⊂
Hλn (p, α). For the ”only if ” part, assume that f ∈ Tλn (p, α). Then by (1.3)
we have
(
(Dλn h(z))′ + (Dλn g(z))′
ℜ
pz p−1
)
=
½
∞
X
¤
k£
1 + λ(k − 1) C(n, k)|ak |z k−p −
=ℜ 1−
p
k=p+1
¾
∞
X
¤
k£
k−p
> α.
−
1 + λ(k − 1) C(n, k)|bk |z
p
k=p
If we choose z to be real and let z → 1− , we get
∞
X
¤
k£
1 + λ(k − 1) C(n, k)|ak |z k−p −
1−
p
k=p+1
∞
X
¤
k£
−
1 + λ(k − 1) C(n, k)|bk |z k−p ≥ α,
p
k=p
which is precisely the assertion (2.4) of Theorem 2.2.
3
Distortion Bounds and Extreme Points.
In this section, we shall obtain distortion bounds for functions in Tλn (p, α)
and also provide extreme points for the class Tλn (p, α).
Theorem 3.1. If f ∈ Tλn (p, α), for λ ≥ 0, p ∈ N, n ∈ N0 and |z| = r > 1,
then
|f (z)| ≤ (1 + bp )rp +
p(1 − α) − |bp |
rp+1 ,
(p + 1)(1 + λ)(n + p)
|f (z)| ≥ (1 − bp )rp −
p(1 − α) − |bp |
rp+1 .
(p + 1)(1 + λ)(n + p)
and
A New Class of Multivalent Harmonic Functions
43
Proof. We only prove the second inequality. The argument for first inequality is similar and will be omitted. Let f ∈ Tλn (p, α). Taking the absolute
value of f , we obtain
p
|f (z)| ≥ (1 − bp )r −
∞
X
(|ak | + |bk |)rk ≥ (1 − bp )rp −
k=p+1
−
∞
X
(|ak | + |bk |)rp+1 =
k=p+1
= (1 − bp )rp −
·
∞
X
1
·
(p + 1)(1 + λ)(n + p)
(p + 1)(1 + λ)(n + p)(|ak | + |bk |)rp+1 ≥
k=p+1
≥ (1 − bp )rp −
·
∞
X
k=p+1
1
·
(p + 1)(1 + λ)(n + p)
£
¤
k 1 + λ(k − p) C(n, k, p)(|ak | + |bk |)rp+1 ≥
≥ (1 − bp )rp −
h
i
1
p(1 − α) − |bp | rp+1 .
(p + 1)(1 + λ)(n + p)
The bounds given in Theorem 3.1 for the functions f = h+g of the form
(1.2) also hold for functions of the form (1.1) if the coefficient condition (2.1)
is satisfied. The functions
f (z) = z p + |bp |z p −
p(1 − α) − |bp |
z p+1
(p + 1)(1 + λ)(n + p)
f (z) = (1 − |bp |)z p −
p(1 − α) − |bp |
z p+1
(p + 1)(1 + λ)(n + p)
and
for |bp | < 1 show that the bounds given Theorem 3.1 are sharp.
44
K. Al Shaqsi and M. Darus
The following covering result follows from the second inequality in The-
orem 3.1.
Corollary 1 If f ∈ Tλn (p, α), then
(
)
p(1 − α) − |bp |
w : |w| < (1 − |bp |) −
⊂ f (U).
(p + 1)(1 + λ)(n + p)
Theorem 3.2. f ∈ Tλn (p, α) if and only if f can be expressed as
(7)
f (z) =
∞
X
(γk hk + µk gk )
k=p
where z ∈ U,
p(1 − α)
¤
hp (z) = z p , hk (z) = z p − £
z k , (k = p + 1, p + 2, ...),
k 1 + λ(k − p) C(n, k, p)
∞
P
p(1 − α)
¤
gk (z) = z p − £
z k , (k = p, p + 1, ...),
k 1 + λ(k − p) C(n, k, p)
(γk + µk ) = 1, γk ≥ 0 and µk ≥ 0 (k = p + 1, p + 2, ...).
k=p
In particular, the extreme points of Tλn (p, α) are {hk } and {gk }.
Proof. Note that for f we may write
f (z) =
∞
X
(γk hk + µk gk ) =
k=p
=
∞
X
k=p
p
(γk + µk )z −
∞
X
p(1 − α)
£
¤
γk z k −
k
1
+
λ(k
−
p)
C(n,
k,
p)
k=p+1
A New Class of Multivalent Harmonic Functions
−
∞
X
k=p
p
=z −
−
45
p(1 − α)
£
¤
µk z k =
k 1 + λ(k − p) C(n, k, p)
∞
X
p(1 − α)
£
¤
γk z k −
k 1 + λ(k − p) C(n, k, p)
k=p+1
∞
X
k=p
Then
p(1 − α)
¤
µk z k
k 1 + λ(k − p) C(n, k, p)
£
∞ h
i
X
£
¤
p(1 − α)
¤
γk
k 1 + λ(k − p) C(n, k, p) £
k
1
+
λ(k
−
p)
C(n,
k,
p)
k=p+1
∞ h
i
X
£
¤
p(1 − α)
¤
µk
k 1 + λ(k − p) C(n, k, p) £
k
1
+
λ(k
−
p)
C(n,
k,
p)
k=p
à ∞
!
X
= p(1 − α)
(γk + µk ) − γp = p(1 − α)(1 − γp ) ≤ p(1 − α)
−
k=p
and so f ∈ Tλn (p, α).
Conversely, suppose that f ∈ Tλn (p, α) . Setting
£
¤
k 1 + λ(k − p) C(n, k, p)
γk =
|ak |(k = p + 1, p + 2, ...),
p(1 − α)
£
¤
k 1 + λ(k − p) C(n, k, p)
µk =
|bk |(k = p, p + 1, p + 2, ...),
p(1 − α)
we obtain
f (z) =
∞
P
(γk hk + µk gk ) as required.
k=p
Acknowledgement: The work presented here was supported by SAGA(Scientific
Advancement Grant Allocation) STGL-012-2006, Academy of Sciences, Malaysia.
46
K. Al Shaqsi and M. Darus
References
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Proc. Amer. Math. Soc. 51, (1998), 283-289.
[2] J. Clunie and T. Shell-Small, Harmonic univalent functions, Ann. Acad.
Aci. Fenn. Ser. A I Math. 9, (1984), 3-25.
[3] K. Al Shaqsi and M. Darus, On univalent functions with respect to
k-symmetric points defined by a generalized Ruscheweyh derivatives
operator.(Submitted)
[4] M. Darus and K.Al Shaqsi, On harmonic univalent functions defined by
a generalized Ruscheweyh derivatives operator, Lobachevskii Journal of
Mathematics 22, (2006), 19-26.
[5] O. Ahuja and J. Jahangiri, Noshiro-type harmonic univalent functions,
Sci. Math. Japon. 65(2), (2002), 293-299.
[6] R. Goel and N.Sohi, New criteria for p-valence, Indian. J. Pure appl.
Math. 7, (2004), 55-61.
[7] S. Yalçin and M. Öztürk, A new subclass of complex harmonic functions, Math. Ineq. Appl. 7, (2004), 55-61.
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi 43600 Selangor D. Ehsan, Malaysia
E-mail address: ommath@hotmail.com
E-mail address: maslina@pkrisc.cc.ukm.my