Observe thatSHreduces toS, the class of normalized univalent analytic functions, if the coan-alytic part of f is zero.. Also, denote by S∗H the subclasses ofSH consisting of functions f
Trang 1Volume 2008, Article ID 263413, 10 pages
doi:10.1155/2008/263413
Research Article
On Harmonic Functions Defined by
Derivative Operator
K Al-Shaqsi and M Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D Ehsan, Malaysia
Correspondence should be addressed to M Darus, maslina@pkrisc.cc.ukm.my
Received 16 September 2007; Revised 20 November 2007; Accepted 26 November 2007
Recommended by Vijay Gupta
Let S H denote the class of functions f h ––g that are harmonic univalent and
sense-preserv-ing in the unit disk U {z : |z| < 1}, where hz z ∞
k2 a k z k , gz ∞k1 b k z k |b1| <
1 In this paper, we introduce the class M Hn, λ, α of functions f h––g which are harmonic in U.
A sufficient coefficient of this class is determined It is shown that this coefficient bound is also
nec-essary for the class M––
Hn, λ, α if f n z h g––n ∈ MHn, λ, α, where hz z −∞
k2 |a k |z k , g n z
−1n∞
k1 |b k |z k and n ∈ N 0 Coefficient conditions, such as distortion bounds, convolution
con-ditions, convex combination, extreme points, and neighborhood for the class M––
Hn, λ, α, are
ob-tained.
Copyright q 2008 K Al-Shaqsi and M Darus This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
A continuous function 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 simply connected domainD ⊂ C, we can write
f h g, where h and g are analytic in D We call h the analytic part and g the coanalytic part
of f A necessary and sufficient condition for f to be locally univalent and sense-preserving in
D is that |hz| > |gz| in D; see 2
Denote bySH the class of functions f h g that are harmonic, univalent, and
sense-preserving in the unit diskU {z : |z| < 1} for which f0 h0 f z0 − 1 0 Then for
f h g ∈ SH, we may express the analytic functions h and g as
hz z ∞
k2
a k z k , gz ∞
k1
b k z k , b1 < 1. 1.1
Trang 2Observe thatSHreduces toS, the class of normalized univalent analytic functions, if the
coan-alytic part of f is zero Also, denote by S∗H the subclasses ofSH consisting of functions f that
mapU onto starlike domain
For f h g given by 1.1, we define the derivative operator introduced by authors
see 1 of f as
Dn
λ fz D n
λ hz −1 nDn
λ gz, n, λ ∈ N0 N ∪ {0}, z ∈ U, 1.2 whereDn
λ hz z ∞
k2 k n Cλ, ka k z k , D n
λ gz ∞
k1 k n Cλ, kb k z k , and Cλ, k kλ−1
We let MHn, λ, α denote the family of harmonic functions f of the form 1.1 such that
ReDn1
Dn
whereDn
λ f is defined by 1.2
If the coanalytic part of f h g is identically zero, then the class MHn, λ, α turns out
to be the classRn
λ α introduced by Al-Shaqsi and Darus 1 for the analytic case
Let MHn, λ, α denote that the subclass of MHn, λ, α consists of harmonic functions
f n h g n such that h and g nare of the form
hz z −∞
k2
a k z k , g n z −1 n∞
k1
It is clear that the class MHn, λ, α includes a variety of well-known subclasses of SH For
example, MH0, 0, α ≡ S∗
Hα is the class of sense-preserving, harmonic, univalent functions
f which are starlike of order α in U, that is, ∂/∂θargfreiθ > α, and MH1, 0, α ≡
MH0, 1, α ≡ HKα is the class of sense-preserving, harmonic, univalent functions f which are convex of order α in U, that is, ∂/∂θarg∂/∂θfreiθ > α Note that the classes
S∗
H and HKα were introduced and studied by Jahangiri 3 Also we notice that the class
MHn, 0, α is the class of Salagean-type harmonic univalent functions introduced by Jahangiri
et al.4; and MH0, λ, α is the class of Ruscheweyh-type harmonic univalent functions
stud-ied by Murugusundaramoorthy and Vijaya5
In 1984, Clunie and Sheil-Small2 investigated the class SHas well as its geometric sub-classes and obtained some coefficient bounds Since then, there has been several related papers
onSHand its subclasses such that Silverman6, Silverman and Silvia 7, and Jahangiri 3,8 studied the harmonic univalent functions Jahangiri and Silverman 9 prove the following theorem
Theorem 1.1 Let f h g given by 1.1 If
∞
k2
then f is sense-preserving, harmonic, and univalent in U and f ∈ S∗
Hconsists of functions inSHwhich are starlike in U.
The condition1.5 is also necessary if f ∈ TH ≡ MH0, 0, 0.
In this paper, we will give sufficient condition for functions f h g, where h and g are given by1.1 to be in the class MHn, λ, α; and it is shown that this coefficient condition is
Trang 3also necessary for functions in the class MHn, λ, α Also, we obtain distortion theorems and characterize the extreme points for functions in MHn, λ, α Closure theorems and application
of neighborhood are also obtained
2 Coefficient bounds
We begin with a sufficient coefficient condition for functions in MHn, λ, α.
Theorem 2.1 Let f h g be given by 1.1 If
∞
k1
k − αa k k αb k k n Cλ, k ≤ 21 − α, 2.1
where a1 1, n, λ ∈ N0, Cλ, k kλ−1
λ , and 0 ≤ α < 1, then f is sense-preserving, harmonic, univalent in U, and f ∈ MHn, λ, α.
Proof If z1/ z2, then
f z1 − f z2
h
z1 − h z2
≥ 1 −g z1 − g z2
h
z1 − h z2
1 −
∞k1 b k
z k
1− z k
2
z1− z2 ∞k2 a k
z k
1− z k
2
>1−
∞
k1 kb k
1−∞k2 ka k
≥1−
∞
k1
k αk n Cλ, k/1 − α k
1−∞k2 k − αk n Cλ, k/1 − α k ≥0,
2.2
which proves univalence Note that f is sense-preserving inU This is because
hz ≥ 1 −∞
k2
ka k |z| k−1
> 1 −∞
k2
k − αk n Cλ, k
1− α a k
≥∞
k1
k αk n Cλ, k
1− α b k
>∞
k1
k αk n Cλ, k
1− α b k |z| k−1≥∞
k1
kb k |z| k−1 ≥gz.
2.3
Using the fact that Rew > α if and only if |1 − α w| ≥ |1 α − w|, it suffices to show that
1 − αD n
λ fz D n1
λ fz − 1 αD n
λ fz − D n1
λ fz ≥ 0. 2.4
Trang 4λ fz in 2.4 yields, by 2.1, we obtain
1 − αD n
λ fz D n1
λ fz − 1 αD n
λ fz − D n1
2− αz
∞
k2
k 1 − αk n Cλ, ka k z k− −1n∞
k1
k − 1 αk n Cλ, kb k z k
−
−αz
∞
k2
k − 1 − αk n Cλ, ka k z k − −1n∞
k1
k 1 αk n Cλ, kb k z k
≥21−α|z| 1−∞
k2
k−αk n Cλ, k
1− α a k |z| k−1∞
k1
kαk n Cλ, k
1− α b k |z| k−1
≥ 21 − α 1−∞
k2
k − αk n Cλ, k
1− α a k −∞
k1
k αk n Cλ, k
1− α b k.
2.5
This last expression is nonnegative by2.1, and so the proof is complete
The harmonic function
fz z ∞
k2
1− α
k − αk n Cλ, k x k z k
∞
k1
1− α
k αk n Cλ, k y k z k , 2.6 where n, λ∈ N0and∞
k2 |x k| ∞k1 |y k| 1 show that the coefficient bound given by 2.1 is sharp The functions of the form2.6 are in MHn, λ, α because
∞
k1
k − α
1− α a k k α
1− α b kk n Cλ, k 1 ∞
k2
x k ∞
k1
In the following theorem, it is shown that the condition 2.1 is also necessary for functions
f n h g n , where h and g nare of the form1.4
Theorem 2.2 Let f n h g n be given by1.4 Then f n ∈ MHn, λ, α if and only if
∞
k1
k − αa k k αb k k n Cλ, k ≤ 21 − α, 2.8
where a1 1, n, λ ∈ N0, Cλ, k kλ−1
λ , and 0 ≤ α < 1.
Proof Since MHn, λ, α ⊂ MHn, λ, α, we only need to prove the “if and only if” part of the theorem To this end, for functions f n of the form1.4, we notice that the condition 1.3 is equivalent to
Re 1 − αz −∞k2 k − αk n Cλ, ka k z k− −12n∞
k1 k αk n Cλ, kb k z k
z −∞k2 k n Cλ, ka k z k −12n∞
k1 k n Cλ, kb k z k
≥ 0. 2.9
Trang 5The above required condition2.9 must hold for all values of z in U Upon choosing the values
of z on the positive real axis, where 0 ≤ z r < 1, we must have
1− α −∞k2 k − αk n Cλ, ka k r k−1−∞k1 k αk n Cλ, kb k r k−1
1−∞k2 k n Cλ, ka k r k−1∞k1 k n Cλ, kb k r k−1 ≥ 0. 2.10
If the condition2.8 does not hold, then the numerator in 2.10 is negative for r sufficiently close to 1 Hence there exist z0 r0 in0, 1 for which the quotient in 2.8 is negative This
contradicts the required condition for f n ∈ MHn, λ, α and so the proof is complete.
3 Distortion bounds
In this section, we will obtain distortion bounds for functions in MHn, λ, α.
Theorem 3.1 Let f n ∈ MHn, λ, α Then for |z| r < 1, one has
f n z ≤ 1 b1
2n λ 1
1− α
2− α −
1 α
2− α b1r2,
f n z ≥ 1 − b1
2n λ 1
1− α
2− α −
1 α
2− α b1r2.
3.1
Proof We only prove the left-hand inequality The proof for the right-hand inequality is similar and will be omitted Let f n ∈ MHn, λ, α Taking the absolute value of f n, we obtain
f n z z −∞
k2
a k z k −1n∞
k1
b k z k
≥ 1−b1
∞
k2
a k b k k
≥ 1−b1 2
∞
k2
a k b k
2 − α2 n λ 1
∞
k2
2 − α2 n λ 1
1− α a k 2 − α2 n λ 1
1− α b kr2
2−α2 n λ1
∞
k2
k−αk n Cλ, k
1− α a k kαk n Cλ, k
1− α b kr2
≥ 1−b1
1− α
2 − α2 n λ 1
1−1 α
1− α b1r2.
3.2 The functions
fz z b1z 1
2n λ 1
1− α
2− α −
1 α
2− α b1z2, fz 1−b1
2n λ 1
1− α
2− α −
1 α
2− α b1z2
3.3 for|b1| ≤ 1 − α/1 α show that the bounds given inTheorem 3.1are sharp
Trang 6The following covering result follows from the left-hand inequality inTheorem 3.1.
Corollary 3.2 If the function f n h g n , where h and g given by 1.4 are in MHn, λ, α, then
w : |w| < 2n1 λ 1 − 1 −
2n λ 1 − 1 α
2n1 λ 1 − 1 − 2n λ 1 1 α
2n λ 12 − α b1⊂f n U.
3.4
4 Convolution, convex combination, and extreme points
In this section, we show that the class MHn, λ, α is invariant under convolution and convex
combination of its member
For harmonic functions f n z z − ∞k2 a k z k −1n∞k1 b k z k and F n z z −
∞
k2 A k z k −1n∞
k1 B k z k , the convolution of f n and F nis given by
f n ∗F n z f n z∗F n z z −∞
k2
a k A k z k −1n∞
k1
Theorem 4.1 For 0 ≤ β ≤ α < 1, let f n ∈ MHn, λ, α and F n ∈ MHn, λ, β Then f n ∗F n
∈ MHn, λ, α ⊂ MHn, λ, β.
Proof We wish to show that the coefficients of f n ∗F n satisfy the required condition given in
function f n ∗F n, we obtain
∞
k2
k − βk n Cλ, k
1− β a k A k ∞
k1
k βk n Cλ, k
1− β b k B k
≤∞
k2
k − βk n Cλ, k
1− β a k ∞
k1
k βk n Cλ, k
1− β b k
≤∞
k2
k − αk n Cλ, k
1− α a k ∞
k1
k αk n Cλ, k
1− α b k ≤ 1,
4.2
since 0≤ β ≤ α < 1 and f n ∈ MHn, λ, α Therefore f n ∗F n ∈ MHn, λ, α ⊂ MHn, λ, β.
We now examine the convex combination of MHn, λ, α.
Let the functions f n j z be defined, for j 1, 2, , by
f n j z z −∞
k2
a k,j z k −1n∞
k1
Theorem 4.2 Let the functions f n j z defined by 4.3 be in the class MHn, λ, α for every j
1, 2, , m Then the functions t j z defined by
t j z m
j1
are also in the class MHn, λ, α, wherem j1 c j 1.
Trang 7Proof According to the definition of t j, we can write
t j z z −∞
k2
j1
c j a k,j
z k −1n∞
k1
j1
c j b n,j
Further, since f n j z are in MHn, λ, α for every j 1, 2, , then by 2.8, we have
∞
k1
k − α
j1
c j a k,j k αm
j1
c j b k,jk n Cλ, k
m
j1
c j
k1
k − αa n,j k αb n,j k n Cλ, k
≤m
j1
c j21 − α ≤ 21 − α
4.6
Hence the theorem follows
Corollary 4.3 The class MHn, λ, α is closed under convex linear combination.
Proof Let the functions f n j z j 1, 2 defined by 4.1 be in the class MHn, λ, α Then the
functionΨz defined by
is in the class MHn, λ, α Also, by taking m 2, t1 μ, and t2 1 − μ inTheorem 4.1, we have the corollary
Next we determine the extreme points of closed convex hulls of MHn, λ, α denoted by clcoMHn, λ, α.
Theorem 4.4 Let f n be given by1.4 Then f n ∈ MHn, λ, α if and only if
f n z ∞
k1
where h1z z, h k z z − 1 − α/k − αk n Cλ, kz k , k 2, 3, , g n k z z
−1n 1−α/k αk n Cλ, kz k , k 1, 2, 3, , and ∞k1 X k Y k 1, X k ≥ 0, Y k ≥ 0 In particular, the extreme points of MHn, λ, α areh k
and
g n k
Proof For the functions f nof the form4.8, we have
f n z ∞
k1
X k h k z Y k g n k z
∞
k1
X k Y k z −∞
k2
1− α
k − αk n Cλ, k X k z k −1n
∞
k1
1− α
k αk n Cλ, k Y k z k .
4.9
Then
∞
k2
k − αk n Cλ, k
1− α a k ∞
k1
k αk n Cλ, k
1− α b k ∞
k2
X k∞
k1
Y k 1 − X1≤ 1, 4.10
Trang 8and so f n ∈ clcoMHn, λ, α.
Conversely, suppose that f n ∈ clcoMHn, λ, α Setting
X k k − αk n Cλ, k
1− α a k , 0≤ X k ≤ 1, k 2, 3, ,
Y k k αk n Cλ, k
1− α b k , 0≤ Y k ≤ 1, k 1, 2, 3, ,
4.11
and X1 1 −∞k2 X k−∞k1 Y k Therefore, f ncan be written as
f n z z −∞
k2
a k z k −1n∞
k1
b k z k
z −∞
k2
1 − αX k
k − αk n Cλ, k z k −1n
∞
k1
1 − αY k
k αk n Cλ, k z k
z ∞
k2
h k z − z X k∞
k1
g n k z − z Y k
∞
k2
h k zX k∞
k1
g n k zY k z
1−∞
k2
X k−∞
k1
Y k
∞
k1
h k zX k g n k zY k , as required.
4.12
Using Corollary 4.3 we have clcoMHn, λ, α MHn, λ, α Then the statement of
5 An application of neighborhood
In this section, we will prove that the functions in a neighborhood of MHn, λ, α are starlike
harmonic functions
Following10, we defined the δ-neighborhood of a function f ∈ TH by
Nδ f Fz z −∞
k2
A k z k−∞
k1
B k z k ,∞
k2
k a k − A k b k − B k b1− B1 ≤ δ, 5.1
where δ > 0.
Theorem 5.1 Let
δ 2 − α2
n λ 1 − 1 α − 2 − α2 n λ 1 − 1 − α 1
ThenNδ MHn, λ, α ⊂ TH.
Trang 9Proof Suppose f n ∈ MHn, λ, α Let F n H G n ∈ Nδ
f n , where H z −∞k2 A k z k and
G n −1n∞k1 B k z k We need to show that F n ∈ TH In other words, it suffices to show that
F nsatisfies the conditionTF ∞k2 k|A k | |B k | |B1| ≤ 1 We observe that
TF ∞
k2
k A k B k B1
∞
k2
k A k − a k a k B k − b k b k B1− b1 b1
∞
k2
k A k − a k B k − b k ∞
k2
k a k b k B1− b1 b1
k2
k A k − a k B k − b k B1− b1∞
k2
k a k b k b1
δ b1 ∞
k2
k a k b k
δ b1 1− α
2 − α2 n λ 1
∞
k2
2− α
1− α a k 2 α
1− α b k2n λ 1
≤ δ b1 1− α
2 − α2 n λ 1
∞
k2
k − α
1− α a k k α
1− α b kk n Cλ, k
≤ δ b1 1− α
2 − α2 n λ 1
1−1 α
1− α b1.
5.3
Now this last expression is never greater than one if
δ ≤ 1 − b1 − 1− α
2 − α2 n λ 1
1−1 α
1− α b1
2 − α2
n λ 1 − 1 α − 2 − α2 n λ 1 − 1 − α 1
5.4
Acknowledgment
The work presented here was supported by Fundamental Research Grant Scheme UKM-ST-01-FRGS0055-2006
References
1 K Al-Shaqsi and M Darus, “An operator defined by convolution involving the polylogarithms func-tions,” submitted.
2 J Clunie and T Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae.
Series A I, vol 9, pp 3–25, 1984.
3 J M Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and
Applications, vol 235, no 2, pp 470–477, 1999.
4 J M Jahangiri, G Murugusundaramoorthy, and K Vijaya, “Salagean-type harmonic univalent
func-tions,” Southwest Journal of Pure and Applied Mathematics, no 2, pp 77–82, 2002.
Trang 105 G Murugusundaramoorthy and K Vijaya, “On certain classes of harmonic functions involving
Ruscheweyh derivatives,” Bulletin of the Calcutta Mathematical Society, vol 96, no 2, pp 99–108, 2004.
6 H Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical
Anal-ysis and Applications, vol 220, no 1, pp 283–289, 1998.
7 H Silverman and E M Silvia, “Subclasses of harmonic univalent functions,” New Zealand Journal of
Mathematics, vol 28, no 2, pp 275–284, 1999.
8 J M Jahangiri, “Coefficient bounds and univalence criteria for harmonic functions with negative co-efficients,” Annales Universitatis Mariae Curie-Skłodowska Sectio A, vol 52, no 2, pp 57–66, 1998.
9 J M Jahangiri and H Silverman, “Harmonic univalent functions with varying arguments,”
Interna-tional Journal of Applied Mathematics, vol 8, no 3, pp 267–275, 2002.
10 S Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical
So-ciety, vol 81, no 4, pp 521–527, 1981.
... 105 G Murugusundaramoorthy and K Vijaya, ? ?On certain classes of harmonic functions involving
Ruscheweyh derivatives,”...
Trang 5The above required condition2.9 must hold for all values of z in U Upon choosing the values
of...
4 Convolution, convex combination, and extreme points
In this section, we show that the class MHn, λ, α is invariant under convolution and convex