Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 190, 2006
INEQUALITIES INVOLVING MULTIPLIERS FOR MULTIVALENT HARMONIC
FUNCTIONS
H. ÖZLEM GÜNEY AND OM P. AHUJA
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE AND A RT
D ICLE U NIVERSITY
D IYARBAKIR 21280, T URKEY
ozlemg@dicle.edu.tr
K ENT S TATE U NIVERSITY
D EPARTMENT OF M ATHEMATICAL S CIENCES
14111, C LARIDON -T ROY ROAD
B URTON , O HIO 44021, U.S.A.
oahuja@kent.edu
Received 17 May, 2006; accepted 30 August, 2006
Communicated by N.E. Cho
A BSTRACT. We introduce inequalities involving multipliers for complex-valued multivalent
harmonic functions, using two sequences of positive real numbers. By specializing those sequences, we determine representation theorems, distortion bounds, integral convolutions, convex combinations and neighborhoods for such functions. The theorems presented, in many cases,
confirm or generalize various well-known results for corresponding classes of multivalent or univalent harmonic functions.
Key words and phrases: Multivalent harmonic, Multivalent harmonic starlike, Multivalent harmonic convex, Multiplier, Integral convolution, Neighborhood.
2000 Mathematics Subject Classification. Primary 30C45; Secondary 30C50.
1. I NTRODUCTION
A continuous complex-valued function f = u + iv defined in a simply connected complex
domain D is said to be harmonic in D if both u and v are real harmonic in D. Such functions
admit the representation f = h + ḡ, where h and g are analytic in D. In [5], it was observed that
f = h + ḡ is locally univalent and sense preserving if and only if |g 0 (z)| < |h0 (z)|, z ∈ D.
The study of harmonic functions which are multivalent in the unit disc U = {z ∈ C : |z| <
1} was initiated by Duren, Hengartner and Laugesen [6]. However, passing from harmonic
univalent functions to the harmonic multivalent functions turns out to be quite non-trivial. In
view of the argument principle for harmonic functions obtained in [6], the second author and
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
The authors wish to thank the referee for suggesting certain improvements in the paper.
145-06
2
H. Ö ZLEM G ÜNEY AND O M P. A HUJA
Jahangiri [1, 2] introduced and studied certain subclasses of the family H(m), m ≥ 1, of all
m-valent harmonic and orientation preserving functions in U. A function f in H(m) can be
expressed as f = h + g, where h and g are analytic functions of the form
(1.1)
m
h (z) = z +
∞
X
an+m−1 z
n+m−1
,
g (z) =
n=2
∞
X
bn+m−1 z n+m−1 ,
|bm | < 1.
n=1
The class H(1) of harmonic univalent functions was studied by Clunie and Sheil-Small [5].
Let SH(m, α), m ≥ 1 and 0 ≤ α < 1 denote the class of functions f = h + g ∈ H(m)
which satisfy the condition
∂
(arg(f (reiθ ))) ≥ mα,
∂θ
(1.2)
for each z = reiθ , 0 ≤ θ < 2π, and 0 ≤ r < 1. A function f in SH(m, α) is called an m-valent
harmonic starlike function of order α. Also, let T H(m, α), m ≥ 1, denote the class of functions
f = h + g ∈ SH(m, α) so that h and g are of the form
(1.3)
m
h (z) = z −
∞
X
|an+m−1 |z
n+m−1
,
n=2
g (z) =
∞
X
|bn+m−1 |z n+m−1 ,
|bm | < 1.
n=1
The class T H(m, α) was studied by second author and Jahangiri in [1, 2]. In particular, they
stated the following:
Theorem A. Let f = h + ḡ be given by (1.3). Then f is in T H(m, α) if and only if
∞ X
n − 1 + m(1 − α)
n − 1 + m(1 + α)
(1.4)
|an+m−1 | +
|bn+m−1 | ≤ 2,
m(1 − α)
m(1 − α)
n=1
where am = 1 and m ≥ 1.
Analogous to T H(m, α) is the class KH(m, α) of m-valent harmonic convex functions of
order α, 0 ≤ α < 1. More precisely, a function f = h + g, where h and g are of the form (1.3),
is in KH(m, α) if and only if it satisfies the condition
∂
∂
iθ
arg
f (re )
≥ mα,
∂θ
∂θ
for each z = reiθ , 0 ≤ θ < 2π, and 0 ≤ r < 1.
Theorem B ([4]). Let f = h + ḡ be given by (1.3). Then f is in KH(m, α) if and only if
(1.5)
∞
X
n+m−1
n=1
m2 (1 − α)
[(n − 1 + m(1 − α))|an+m−1 | + (n − 1 + m(1 + α))|bn+m−1 | ] ≤ 2,
where am = 1 and m ≥ 1.
Inequalities (1.4) and (1.5) as well as several such known inequalities in the literature are the
motivating forces for introducing a multiplier family Fm ({cn+m−1 }, {dn+m−1 }) for m ≥ 1. A
function f = h + ḡ, where h and g are given by (1.3), is said to be in the multiplier family
Fm ({cn+m−1 }, {dn+m−1 }) if there exist sequences {cn+m−1 } and {dn+m−1 } of positive real
numbers such that
∞ X
cn+m−1
dn+m−1
(1.6)
|an+m−1 | +
|bn+m−1 | ≤ 2,
cm = m, dm |bm | < m.
m
m
n=1
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
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I NEQUALITIES I NVOLVING M ULTIPLIERS F OR M ULTIVALENT H ARMONIC F UNCTIONS
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The multipliers {cn+m−1 } and {dn+m−1 } provide a transition from multivalent harmonic starlike functions to multivalent harmonic convex functions, including many more subclasses of
H(m) and H(1). For example,
n − 1 + m(1 − α)
n − 1 + m(1 + α)
(1.7)
Fm
,
≡ T H(m, α),
1−α
1−α
(1.8) Fm
(n + m − 1)(n − 1 + m(1 − α))
,
m(1 − α)
(n + m − 1)(n − 1 + m(1 + α))
≡ KH(m, α),
m(1 − α)
Fm ({n + m − 1}, {n + m − 1}) ≡ T H(m, 0) := T H(m),
(1.9)
(1.10)
Fm
(n + m − 1)2
m
(n + m − 1)2
,
≡ KH(m, 0) := KH(m),
m
(1.11)
F1 ({n}, {n}) ≡ T H(1, 0) = T H,
(1.12)
F1 ({n2 }, {n2 }) ≡ KH(1, 0) := KH,
(1.13)
F1 ({np }, {np }) := F ({np }, {np }),
p > 0,
F1 ({cn }, {dn }) := F ({cn }, {dn }).
(1.14)
While (1.7), (1.9) and (1.11) follow immediately from Theorem A, (1.8), (1.10) and (1.12)
are consequences of Theorem B. Note that T H and KH in (1.11) and (1.12) were studied in
[9] as well as [10]. Also, by letting m = 1, α = 0, cn = dn = np for p > 0 and b1 = 0, the
classes F1 ({np }, {np }) were studied in [8]. Finally, (1.14) follows from (1.6) by setting m = 1
which was studied in [3].
In this paper, we determine representation theorems, distortion bounds, convolutions, convex
combinations and neighborhoods of functions in Fm ({cn+m−1 }, {dn+m−1 }). As illustrations of
our results, the corresponding results for certain families are presented in the corollaries.
2. M AIN R ESULTS
If (n + m − 1) ≤ cn+m−1 and (n + m − 1) ≤ dn+m−1 , then by Theorem A we have
Fm ({cn+m−1 }, {dn+m−1 }) ⊂ T H(m).
Consequently, the functions Fm ({cn+m−1 }, {dn+m−1 }) are sense-preserving, harmonic and multivalent in U. We first observe that if
∞
∞
X
X
|b1(n+m−1) |z̄ n+m−1
|a1(n+m−1) |z n+m−1 +
f1 (z) = z m −
n=1
n=2
and
m
f2 (z) = z −
∞
X
|a2(n+m−1) |z
n=2
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
n+m−1
+
∞
X
|b2(n+m−1) |z̄ n+m−1
n=1
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4
H. Ö ZLEM G ÜNEY AND O M P. A HUJA
are in Fm ({cn+m−1 }, {dn+m−1 }) and 0 ≤ λ ≤ 1, then so is the linear combination λf1 + (1 −
λ)f2 by (1.6). Therefore, Fm ({cn+m−1 }, {dn+m−1 }) is a convex family.
Next we determine the extreme points of the closed convex hull of the family Fm ({cn+m−1 },
{dn+m−1 }), denoted by clcoFm ({cn+m−1 }, {dn+m−1 }).
Theorem 2.1. A function f = h + g is in clcoFm ({cn+m−1 }, {dn+m−1 }) if and only if f has the
representation
∞
X
(2.1)
f (z) =
(λn+m−1 hn+m−1 (z) + µn+m−1 gn+m−1 (z)),
n=1
where
(n + m − 1) ≤ cn+m−1 , (n + m − 1) ≤ dn+m−1 , λn+m−1 ≥ 0,
∞
X
µn+m−1 ≥ 0,
(λn+m−1 + µn+m−1 ) = 1, hm (z) = z m ,
n=1
m
z n+m−1
m
z̄ n+m−1 .
cn+m−1
dn+m−1
In particular, the extreme points of Fm ({cn+m−1 }, {dn+m−1 }) are {hn+m−1 }, {gn+m−1 }.
hn+m−1 (z) = z m −
gn+m−1 (z) = z m +
and
Proof. For functions f of the form (2.1) we have
∞
X
f (z) = λm hm (z) +
λn+m−1 z m −
n=2
∞
X
+
n=1
m
=z −
∞
X
λn+m−1
n=2
m
cn+m−1
z
cn+m−1
m
m
µn+m−1 z +
m
dn+m−1
n+m−1
+
z̄
z
n+m−1
n+m−1
∞
X
µn+m−1
n=1
m
dn+m−1
z̄ n+m−1 .
Then
∞
X
n=2
λn+m−1
m
cn+m−1
cn+m−1 +
∞
X
µn+m−1
n=1
m
dn+m−1
=
∞
X
dn+m−1
mλn+m−1 +
n=2
∞
X
mµn+m−1 ≤ m,
n=1
and so f ∈ clcoFm ({cn+m−1 }, {dn+m−1 }). Conversely, suppose f ∈ clcoFm ({cn+m−1 }, {dn+m−1 }).
We set
cn+m−1
λn+m−1 =
|an+m−1 |,
(n = 2, 3, . . . ),
m
dn+m−1
µn+m−1 =
|bn+m−1 |,
(n = 1, 2, 3, . . . ),
m
and
∞
∞
X
X
λm = 1 −
λn+m−1 −
µn+m−1 .
n=2
n=1
Therefore, by using routine computations, f can be written as
∞
X
f (z) =
(λn+m−1 hn+m−1 (z) + µn+m−1 gn+m−1 (z)).
n=1
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
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I NEQUALITIES I NVOLVING M ULTIPLIERS F OR M ULTIVALENT H ARMONIC F UNCTIONS
5
In view of (1.7), Theorem 2.1 yields:
Corollary 2.2 ([2]). A function f = h + g is in clcoT H(m, α) if and only if f can be expressed
in the form (2.1), where
hm (z) = z m ,
hn+m−1 (z) = z m −
gn+m−1 (z) = z m +
and
∞
X
m(1 − α)
z n+m−1 ,
n − 1 + m(1 − α)
m(1 − α)
z n+m−1 ,
n − 1 + m(1 + α)
(λn+m−1 + µn+m−1 ) = 1, λn+m−1 ≥ 0,
(n = 2, 3, . . . ),
(n = 1, 2, 3, . . . )
µn+m−1 ≥ 0.
n=1
Our next result provides distortion bounds for the functions in Fm ({cn+m−1 }, {dn+m−1 }).
Theorem 2.3. Let {cn+m−1 } and {dn+m−1 } be increasing sequences of positive real numbers
so that
cm+1 ≤ dm+1 ,
(n + m − 1) ≤ cn+m−1
and
(n + m − 1) ≤ dn+m−1
for all n ≥ 2. If f ∈ Fm ({cn+m−1 }, {dn+m−1 }), then
m − dm |bm | m+1
m − dm |bm | m+1
m
m
r
≤ |f (z)| ≤ (1 + |bm |)r +
r
.
(1 − |bm |)r −
cm+1
cm+1
The bounds given above are sharp for the functions
m − dm |bm |
m
m
f (z) = z ± |bm |z̄ +
z̄ m+1 ,
cm+1
dm |bm | < 1.
Proof. Let f ∈ Fm ({cn+m−1 }, {dn+m−1 }). Taking the absolute value of f , we obtain
∞
∞
X
X
m
n+m−1
n+m−1 |f (z) | = z −
|an+m−1 |z
+
|bn+m−1 |z̄
≤ rm +
n=2
∞
X
|an+m−1 |rn+m−1 +
n=2
n=1
∞
X
|bn+m−1 |rn+m−1
n=1
= (1 + |bm |)rm +
∞
X
(|an+m−1 | + |bn+m−1 |)rn+m−1
n=2
1
m
≤ (1 + |bm |)r +
≤ (1 + |bm |)rm +
≤ (1 + |bm |)rm +
≤ (1 + |bm |)rm +
cm+1
1
cm+1
∞
X
n=2
∞
X
1
n=2
∞
X
cm+1
n=2
1
cm+1
cm+1 (|an+m−1 | + |bn+m−1 |)rm+1
(cm+1 |an+m−1 | + dm+1 |bn+m−1 |)rm+1
(cn+m−1 |an+m−1 | + dn+m−1 |bn+m−1 |)rm+1
(m − dm |bm |)rm+1 .
We omit the proof of the left side of the inequality as it is similar to that of the right side.
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
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6
H. Ö ZLEM G ÜNEY AND O M P. A HUJA
Corollary 2.4. If f ∈ T H(m, α), then
|f (z)| ≤ (1 + |bm |)rm +
m(1 − α − (1 + α)|bm |) m+1
r
,
1 + m(1 − α)
|f (z)| ≥ (1 − |bm |)rm −
m(1 − α − (1 + α)|bm |) m+1
r
,
1 + m(1 − α)
and
where |z| = r < 1.
The following covering result follows from the left hand inequality in Theorem 2.3.
Corollary 2.5. Let f be as in Theorem 2.3. Then
1
(cm+1 − m − (cm+1 − dm )|bm |) ⊂ f (U).
w : |w| <
cm+1
Corollary 2.6. If f ∈ T H(m, α), then
1 + (2mα − 1)|bm |
w : |w| <
⊂ f (U).
1 + m(1 − α)
Remark 2.7. For α = 0, the corresponding results in Corollary 2.4 and Corollary 2.6 were also
found in [1].
In the next result, we find the convex combinations of the members of the family Fm ({cn+m−1 },
{dn+m−1 }).
Theorem 2.8. If (n + m − 1) ≤ cn+m−1 and (n + m − 1) ≤ dn+m−1 for all n + m − 1 ≥ 2,
then Fm ({cn+m−1 }, {dn+m−1 }) is closed under convex combinations.
Proof. Consider
m
fi (z) = z −
∞
X
|ain+m−1 |z
n+m−1
+
n=2
∞
X
|bin+m−1 |z̄ n+m−1
n=1
for i = 1, 2, . . . . If fi ∈ Fm ({cn+m−1 }, {dn+m−1 }) then
∞
∞
X
X
(2.2)
cn+m−1 |ain+m−1 | +
dn+m−1 |bin+m−1 | ≤ m,
For
P∞
i=1 ti
∞
X
n=2
n=1
= 1,
0 ≤ ti ≤ 1, we have
ti fi (z) = z m −
∞
∞
X
X
n=2
i=1
!
ti |ain+m−1 | z n+m−1 +
∞
∞
X
X
n=1
i=1
i = 1, 2, . . . .
!
ti |bin+m−1 | z̄ n+m−1 .
i=1
In view of the above equality and (2.2), we obtain
∞
∞
∞
∞
X
X
X
X
cn+m−1 ti ain+m−1 +
dn+m−1 ti |bin+m−1 |
n=2
n=1
i=1
i=1
)
(
∞
∞
∞
X
X
X
=
ti
cn+m−1 |ain+m−1 | +
dn+m−1 |bin+m−1 |
i=1
≤
∞
X
n=2
n=1
ti m = m.
i=1
Hence
P∞
i=1 ti fi
∈ Fm ({cn+m−1 }, {dn+m−1 }), by an application of (1.6).
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
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I NEQUALITIES I NVOLVING M ULTIPLIERS F OR M ULTIVALENT H ARMONIC F UNCTIONS
7
In view of relations (1.7) to (1.10), we have the following results:
Corollary 2.9. The family T H(m, α), KH(m, α), T H(m) and KH(m) are closed under convex combinations.
For harmonic functions
∞
X
f (z) = z m −
(2.3)
|an+m−1 |z n+m−1 +
n=2
∞
X
|bn+m−1 |z̄ n+m−1
n=1
and
m
F (z) = z −
(2.4)
∞
X
|An+m−1 |z
n+m−1
n=2
+
∞
X
|Bn+m−1 |z̄ n+m−1
n=1
define the integral convolution of f and F as
(2.5)
m
(f F ) (z) = z −
∞
X
|an+m−1 An+m−1 |
n+m−1
n=2
z
n+m−1
+
∞
X
|bn+m−1 Bn+m−1 |
n=1
n+m−1
z̄ n+m−1 .
In the following result, we show the integral convolution property of the class Fm ({cn+m−1 },
{dn+m−1 }).
Theorem 2.10. Let (n + m − 1) ≤ cn+m−1 and (n + m − 1) ≤ dn+m−1 for all n + m − 1 ≥ 2.
If f and F are in Fm ({cn+m−1 }, {dn+m−1 }), then so is f F .
Proof. Since Fm ({cn+m−1 }, {dn+m−1 }) ⊂ T H(m) and F ∈ Fm ({cn+m−1 }, {dn+m−1 }), it follows that |An+m−1 | ≤ 1 and |Bn+m−1 | ≤ 1. Then f F ∈ Fm ({cn+m−1 }, {dn+m−1 }) because
∞
X
n=2
≤
∞
X
cn+m−1
dn+m−1
|an+m−1 An+m−1 | +
|bn+m−1 Bn+m−1 |
m(n + m − 1)
m(n
+
m
−
1)
n=1
∞
X
n=2
≤
∞
X
cn+m−1
dn+m−1
|an+m−1 | +
|bn+m−1 |
m(n + m − 1)
m(n
+
m
−
1)
n=1
∞
X
cn+m−1
n=2
m
|an+m−1 | +
∞
X
dn+m−1
n=1
m
|bn+m−1 |
≤ 2.
Corollary 2.11. If f and F are in T H(m, α), KH(m, α), T H(m) and KH(m), then so is
f F.
The δ−neighborhood of the functions f = h + ḡ in Fm ({(n + m − 1)cn+m−1 }, {(n + m −
1)dn+m−1 }) is defined as the set Nδ (f ) consisting of functions
m
m
F (z) = z + Bm z̄ +
∞
X
(An+m−1 z n+m−1 + Bn+m−1 z̄ n+m−1 )
n=2
such that
∞
X
[(n + m − 1)(|an+m−1 − An+m−1 | + |bn+m−1 − Bn+m−1 |)]
n=2
+ m|bm − Bm | ≤ δ,
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
δ > 0.
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8
H. Ö ZLEM G ÜNEY AND O M P. A HUJA
Our next result guarantees that the functions in a neighborhood of
Fm ({(n + m − 1)cn+m−1 }, {(n + m − 1)dn+m−1 })
are multivalent harmonic starlike functions.
Theorem 2.12. Let {cn+m−1 } and {dn+m−1 } be increasing sequences of real numbers so that
cm+1 ≤ dm+1 , (n + m − 1) ≤ cn+m−1 and (n + m − 1) ≤ dn+m−1 for all n ≥ 2. If
m
(cm+1 − 1 − (cm+1 − dm )|bm |),
δ=
cm+1
then
Nδ (Fm ({(n + m − 1)cn+m−1 }, {(n + m − 1)dn+m−1 })) ⊂ T H(m).
Proof. Suppose
f = h + ḡ ∈ Fm ({(n + m − 1)cn+m−1 }, {(n + m − 1)dn+m−1 }).
Let F = H + Ḡ ∈ Nδ (f ) where
m
H(z) = z +
∞
X
An+m−1 z
n+m−1
G(z) =
and
n=2
∞
X
Bn+m−1 z n+m−1 .
n=1
We need to show that F ∈ T H(m). It suffices to show that F satisfies the condition
M (F ) :=
∞
X
(n + m − 1)(|An+m−1 | + |Bn+m−1 |) + m|Bm | ≤ m
n=2
Note that
M (F ) ≤
∞
X
(n + m − 1)[|An+m−1 − an+m−1 | + |Bn+m−1 − bn+m−1 |] + m|Bm − bm |
n=2
+
∞
X
(n + m − 1)(|an+m−1 | + |bn+m−1 |) + m|bm |
n=2
≤ δ + m|bm | +
∞
X
(n + m − 1)(|an+m−1 | + |bn+m−1 |)
n=2
= δ + m|bm | +
1
∞
X
cm+1
n=2
cm+1 (n + m − 1)|an+m−1 |
+ cm+1 (n + m − 1)|bn+m−1 |
∞
1 X
≤ δ + m|bm | +
(n + m − 1)cn+m−1 |an+m−1 |
cm+1 n=2
+ (n + m − 1)dn+m−1 |bn+m−1 |
1
≤ δ + m|bm | +
(m(1 − dm |bm |)).
cm+1
But, the last expression is never greater than m provided that
δ ≤ m − m|bm | −
1
cm+1
(m(1 − dm |bm |)) =
m
cm+1
(cm+1 − 1 − (cm+1 − dm )|bm |).
J. Inequal. Pure and Appl. Math., 7(5) Art. 190, 2006
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I NEQUALITIES I NVOLVING M ULTIPLIERS F OR M ULTIVALENT H ARMONIC F UNCTIONS
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Corollary 2.13. If
δ=
m − (m − 2m2 α)|bm |
,
1 + m(1 − α)
then Nδ (KH(m, α)) ⊂ T H(m).
Letting α = 0 and m = 1, Corollary 2.13 yields the following interesting result.
Corollary 2.14. N 1 (1−|b1 |) (KH) ⊂ T H.
2
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http://jipam.vu.edu.au/