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In this paper, we introduce a class of meromorphic harmonic function with respect to k-symmetric points defined byDj m.. Coeﬃcient bounds, distortion theorems, extreme points, convolutio

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Volume 2008, Article ID 259205, 11 pages

doi:10.1155/2008/259205

Research Article

On Meromorphic Harmonic Functions with

K Al-Shaqsi and M Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor D Ehsan 43600, Malaysia

Correspondence should be addressed to M Darus,maslina@ukm.my

Received 22 May 2008; Revised 20 July 2008; Accepted 23 August 2008

Recommended by Ramm Mohapatra

In our previous work in this journal in 2008, we introduced the generalized derivative operator

Dj m for f ∈ SH In this paper, we introduce a class of meromorphic harmonic function with

respect to k-symmetric points defined byDj m Coeﬃcient bounds, distortion theorems, extreme points, convolution conditions, and convex combinations for the functions belonging to this class are obtained

Copyrightq 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 domain D ⊂ C

if both u and v are real harmonic in D In any simply connected domain, we write f h g where h and g are analytic in D A necessary and suﬃcient condition for f to be locally univalent and orientation preserving in D is that |h| > |g| in D see 1 Hengartner and Schober2 investigated functions harmonic in the exterior of the unit disk U {z : |z| > 1} They showed that complex valued, harmonic, sense preserving, univalent mapping f must

admit the representation

where hz and gz are defined by

n1

b n z −n , 1.2 for 0≤ |β| < |α|, A ∈ C and z ∈ U.

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2 Journal of Inequalities and Applications

For z ∈ U \ {0}, let MHdenote the class of functions:

n1

a n z n∞

n1

b n z n , 1.3

which are harmonic in the punctured unit diskU \ {0}, where hz and gz are analytic in

U \ {0} and U, respectively, and hz has a simple pole at the origin with residue 1 here.

In 3, the authors introduced the operator Dj m for f ∈ SH which is the class of

functions f h g that are harmonic univalent and sense-preserving in the unit disk

U {z : |z| < 1} for which f0 h0 f z0 − 1 0 For more details about the operator

Dj m, see4

Now, we defineDj m for f h g given by 1.3 as

where

n1

Cm, n

n m − 1 m

n m − 1!

1.5

A function f ∈ MHis said to be in the subclassMS∗

Hof meromorphically harmonic starlike functions inU \ {0} if it satisfies the condition

Re

−zhz − zgz

hz gz

Note that the class of harmonic meromorphic starlike functions has been studied by Jahangiri and Silverman5, and Jahangiri 6

Now, we have the following definition

meromorphic harmonic functions f of the form1.3 such that

Re

−D

j1

m fz

Dj m f k z

where

Dj m f k z D j m h k −1jDj m g k j, m ∈ N0, k ≥ 1, 1.8

h k z −1j

n1

a nΦn z n , g k z ∞

n1

Φn z n , 1.9

Φn 1kk−1

ν0

ε n−1ν ,

k ≥ 1; ε exp

2πi

k

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For more details about harmonic functions with respect to k-symmetric points, see

papers7,8 given by the authors

Also, note thatMHS2s j, 0, α ⊂ MHS∗

S n, α was introduced by Bostancı and ¨Ozt ¨urk

9

Finally, letMHSs k j, m, α denote the subclass of MHS s k j, m, α consist of harmonic functions f j h j g j such that h j and g jare of the form

h j z −1z j ∞

n1

|a n |z n , g j z −1 j∞

n1

|b n |z n 1.11

Also, let f k j h k j g k j where h k j and g k jare of the form

h k j z −1j

n1

Φn |a n |z n , g k j z −1 j∞

n1

Φn |b n |z n , 1.12

whereΦnis given by1.10

In this paper, we will give a suﬃcient condition for functions f h g, where h and

condition is also necessary for functions to be in the classMHSk s j, m, α Also, we obtain

distortion bounds and characterize the extreme points for functions in MHSk s j, m, α.

Convolution and closure theorems are also obtained

2 Coefficient bounds

First, we prove a suﬃcient coeﬃcient bound

Theorem 2.1 Let f h g be of the form 1.3 and f k h k g k where h k and g k are given by

1.9 If

∞

n1

n − 1k 1 α|a n−1k1 | n − 1k 1 − α|b n−1k1|Ωj m n, k

∞

n2

n / lk1

2.1

f z1 − f z2

≥h z1 − h z2 1 − g z2

≥ z1− z2

z1z2 −z1− z2∞

n1

a n b n n−1

1 · · · z n−1

2

> z1− z2

z1z21−z22∞

n1

n a n b n

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> z1− z2

z1z2 1−z22

∞

n1

n a n b n

∞

n1

n − 1k 1 a n−1k1 b n−1k1

> z1− z2

z1z21−∞

n1

n − 1k 1 αa n−1k1 − n − 1k 1 − αb n−1k1Ωj

m n, k

− ∞

n2

n / lk1

n j1 Cm, n a n b n .

2.2 This last expression is nonnegative by2.1, and so f is univalent in U \ {0} To show that f

is sense preserving inU \ {0}, we need to show that |hz| ≥ |gz| in U \ {0} We have

|hz| ≥ 1

|z|2 −∞

n1

n|a n ||z| n−1

1

n1

n|a n |r n−1 > 1 −∞

n1

n|a n|

≥ 1 −∞

n1

n − 1k 1 α|a n−1k1|Ωj m n, k − ∞

n2

n / lk1

≥∞

n1

n − 1k 1 − α|b n−1k1|Ωj m n, k ∞

n2

n / lk1

≥∞

n1

2n|b 2n| ∞

n1

2n − 1|b 2n−1|

>∞

n1

n|b n ||z| n−1 ≥ |gz|.

2.3

Now, we will show that f ∈ MHSs k j, m, α According to 1.4 and 1.7, for 0 ≤ α < 1, we

have

Re

−D

j1

m fz

Dj m f k z

Re

⎧

⎨

⎩−

Dj1 m hz − −1 jDj1 m gz

Dj m h k z −1 jDj m g k z

⎫

⎬

⎭ ≥α. 2.4 Using the fact that Re{w} ≥ α if and only if |1 − α w| ≥ |1 α − w|, it suﬃces to show that

1 − α −Dj1 m fz

Dj m f k z

≥1 α Dj1 m fz

Dj m f k z

which is equivalent to

Dj1

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SubstitutingDj m fz, D j1 m fz, and D j m f k z in 2.6 yields

Dj1

−

Dj1

−1

j

n1

1 − α −1z j ∞

n1

n j Cm, nΦ n b n z n

−

−1

j

n1

− 1 α −1z j ∞

n1

n j Cm, nΦ n b n z n

2 − α−1

j

n1

−

j

n1

≥ 2 − α |z| −∞

n1

−|z| α −∞

n1

21 − α|z|

1−∞

n1

1− α |a n|zn1 −∞

n1

1− α |b n|zn1

≥ 21 − α

1−∞

n1

1− α |a n| −

∞

n1

1− α |b n|

.

2.7 From the definition ofΦn , we know that

Φn

⎧

⎨

⎩

1, n lk 1,

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Substituting2.8 in 2.7, then 2.7 is equivalent to

Dj1

≥ 21 − α

1−∞

n1

nk 1 j Cm, nk 1nk 1 α

1− α |a nk1|

−∞

n1

nk 1 j Cm, nk 1nk 1 − α

1− α |b nk1| −

∞

n2

n / lk1

1− α |a n|

− ∞

n2

n / lk1

1− α |b n| −

1 α

1− α |a1| − |b1|

21 − α

1−∞

n1

n − 1k 1 α

1− α |a n−1k1| −

n − 1k 1 − α

1− α |b n−1k1|

Ωj m n, k

− ∞

n2

n / lk1

1− α |a n | |b n|

≥ 0, by 2.6.

2.9 Thus, this completes the proof of the theorem

We next show that condition2.1 is also necessary for functions in MHSs k j, m, α.

Theorem 2.2 Let f j h j g j , where h j and g j are given by1.11, and f k j h k j g k j where h k j

2.1 does not hold We note that for f j ∈ MHSs k j, m, α, then by 1.7 the condition 2.4

must be satisfied for all values of z in U \ {0} Substituting for h j , g j , h k j , and g k j given by

1.11 and 1.12, respectively, in 2.4 and choosing 0 < z r < 1, we are required to have

Re{Ψz/Υz} ≥ 0, where

Ψz −D j1 m h j z −1 nDj1 m g j z − αD j m h k j z − α−1 jDj m g k j z

1− α z −∞

n1

Υz D j m h k j z −1 jDj m g k j z

z1 ∞

n1

n j Cm, nΦ n |b n |z n

2.10

Then, the required condition Re{Ψz/Υz} ≥ 0 is equivalent to

1 − α/z −∞

1/z∞

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By using2.8, and if condition 2.1 does not hold, then the numerator of 2.11 is negative

for r suﬃciently close to 1 Thus, there exists a z0 r0in0, 1 for which the quotient in 2.11

is negative This contradicts the required condition for f j ∈ MHSk s j, m, α and so the proof

is complete

3 Distortion bounds and extreme points

In this section, we will obtain distortion bounds for functions f j ∈ MHSk s j, m, α and also

provide extreme points for the classMHSk s j, m, α.

Theorem 3.1 If f j h j g j∈ MHSs k j, m, α and 0 < |z| r < 1, then

1

2j m 12 − α r ≤ |f j z| ≤

1

2j m 12 − α r. 3.1

Proof We will prove the left side of the inequality The argument for the right side of the

inequality is similar to the left side, and thus the details will be omitted Let f j h j g j ∈ MHSs k j, m, α Taking the absolute value of f, we obtain

|f j|

−1

j

n1

a n z n −1n∞

n1

b n z n

≥ 1r −∞

n1

|a n | |b n |r n

≥ 1r −∞

n1

|a n | |b n |r

≥ 1

2j m 12 − αΦ2

∞

n1

2j m 12 − αΦ2

1− α |a n | |b n |r

≥ 1

2j m 12 − α

∞

n1

n

1− α |a n|

1− α |b n|

r

≥ 1r − 1− α

2j m 12 − α r, by 2.7.

3.2

The bounds given inTheorem 3.1hold for functions f j h g j of the form1.11 And it is also discovered that the bounds hold for functions of the form1.3, if the coeﬃcient condition

2.1 is satisfied

The following covering result follows from the left-hand side of the inequality in

Theorem 3.1

Corollary 3.2 If f j∈ MHSk s j, m, α, then

f jU \ {0} ⊂

2j m 12 − α

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Next, we determine the extreme points of closed convex hulls of MHSk s j, m, α

denoted by clcoMHSs k j, m, α.

Theorem 3.3 Let f j h j g j where h j and g j are given by1.11 Then, f j ∈ MHSs k j, m, α if

and only if

f j,n z ∞

n0

x n h j n z y n g j n z, 3.4

1, 2, 3, , g j,n z −1 j /z−1 j 1−α/n j Cm, nn−αΦ n z k n 1, 2, 3, , ∞

n0 x n

y n 1, x n ≥ 0, y n ≥ 0 In particular, the extreme points of MHS k s j, m, α are {h j,n } and {g j,n }.

f j,n z ∞

n0

x n h j,n z y n g j,n z

∞

n0

x n y n−1z j ∞

n1

1− α

−1j∞

n1

1− α

3.5

Now, the first part of the proof is complete, andTheorem 2.2gives

∞

n1

1− α

1− α x n

∞

n1

1− α

1− α y n

∞

n0

x n y n − x0 y0 1 − x0 y0 ≤ 1.

3.6

Conversely, suppose that f j∈ clcoMHSk s j, m, α For n 1, 2, 3, , set

1− α |a n | 0 ≤ x n ≤ 1,

1− α |b n | 0 ≤ y n ≤ 1,

3.7

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x0 1 −∞n1 x n y n Therefore, f can be written as

f j,n z −1z j ∞

n1

|a n |z n −1j∞

n1

|b n |z n

−1j

n1

1 − αx n

j∞

n1

1 − αy n

n

−1z j ∞

n1

h j,n z −−1z j

n1

g j,n z −−1z j

y n

∞

n1

g j,n zy n −1z j

1−∞

n1

y n

∞

n0

h j,n zx n g j,n zy n , as required.

3.8

4 Convolution and convex combination

In this section, we show that the classMHSs k j, m, α is invariant under convolution and

convex combination of its member

For harmonic functions f j z −1 j /z ∞

n1 |a n |z n −1j∞

n1 |b n |z n and F j z

−1j /z ∞

n1 |A n |z n −1j∞

n1 |B n |z n , the convolution of f j and F jis given by

f j ∗F j z f j z∗F j z −1z j ∞

n1

|a n ||A n |z n −1j∞

n1

|b n ||B n |z n 4.1

Theorem 4.1 For 0 ≤ β ≤ α < 1, let f j ∈ MHSs k j, m, α and F j ∈ MHSk s j, m, β Then,

f j ∗F j∈ MHSk s j, m, α ⊂ MHS s k j, m, β.

Theorem 2.2 For F j ∈ MHSk s j, m, β, we note that |A n | ≤ 1 and |B n| ≤ 1 Now, for the

convolution function f j ∗F j, we obtain

∞

n1

1− β |a n ||A n|

∞

n1

1− β |b n ||B n|

≤∞

n1

1− β |a n|

∞

n1

1− β |b n|

≤∞

n1

1− α |a n|

∞

n1

1− α |b n | ≤ 1,

4.2

since 0 ≤ β ≤ α < 1 and f j ∈ MHSk s j, m, α Therefore f j ∗F j ∈ MHSs k j, m, α ⊂

MHSs k j, m, β.

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We now examine the convex combination ofMHSk s j, m, α.

Let the functions f j,t be defined, for t 1, 2, , ρ, by

f j,t z −1z j ∞

n1

|a n,t |z n −1j∞

n1

|b n,t |z n 4.3

Theorem 4.2 Let the functions f j,t defined by4.3 be in the class MHS k s j, m, α for every t

1, 2, , ρ Then, the functions ξ t z defined by

ξ t z

ρ

t1

c t f j n z, 0 ≤ c t ≤ 1, 4.4

ξ t z −1z j ∞

n1

ρ

t1

c t a n,t

z n −1j∞

n1

ρ

t1

c t b n,t

z n 4.5

Further, since f j,t z are in MHS s k j, m, α for every t 1, 2, , ρ Then by 2.7, we have

∞

n1

n αΦ n

ρ

t1

c t |a n,t|

n − αΦ n

ρ

t1

c t |b n,t|

ρ

t1

c t

∞

n1

n αΦ n |a n,t | n − αΦ n |b n,t |n j Cm, n

≤

ρ

t1

c t 1 − α ≤ 1 − α.

4.6

Hence, the theorem follows

Corollary 4.3 The class MHS s k j, m, α is close under convex linear combination.

the function ψz defined by

is in the classMHSk s j, m, α Also, by taking ρ 2, ξ1 μ, and ξ2 1 − μ inTheorem 4.2,

we have the corollary

Acknowledgment

The work here was fully supported by Fundamental Research GrantSAGA: STGL-012-2006, Academy of Sciences, Malaysia

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8 K Al-Shaqsi and M Darus, ? ?On harmonic univalent functions. ..

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We now examine the convex combination ofMHSk... 7

By using2.8, and if condition 2.1 does not hold, then the numerator of 2.11 is negative

for r suﬃciently close to Thus, there exists

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