On Meromorphic Harmonic Functions with Respect to Symmetric Points

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Journal of Inequalities and Applications Volume 2008, Article ID 259205,11pages doi:10.1155/2008/259205

Research Article

On Meromorphic Harmonic Functions with

Respect to

k

-Symmetric Points

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,[email protected]

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

Djm forf ∈ SH. In this paper, we introduce a class of meromorphic harmonic function with respect tok-symmetric points defined byDjm. Coeﬃcient bounds, distortion theorems, extreme

points, convolution conditions, and convex combinations for the functions belonging to this class are obtained.

Copyrightq2008 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 functionf uivis a complex valued harmonic function in a domainD⊂C if bothuandvare real harmonic inD. In any simply connected domain, we writef hg where hand g are analytic in D. A necessary and suﬃcient condition for f to be locally univalent and orientation preserving inD is that|h| > |g|inD see1. Hengartner and

Schober2investigated functions harmonic in the exterior of the unit diskU {z:|z|>1}. They showed that complex valued, harmonic, sense preserving, univalent mappingfmust admit the representation

fz hz gz Alog|z|, 1.1

wherehzandgzare defined by

hz αz∞

n1

anz−n, gz βz

∞

n1

bnz−n, 1.2

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Forz∈U\ {0},letMHdenote the class of functions:

fz hz gz 1

z

∞

n1

anzn

∞

n1

bnzn, 1.3

which are harmonic in the punctured unit diskU\ {0}, wherehzandgzare analytic in U\ {0}andU, respectively, andhzhas a simple pole at the origin with residue 1 here.

In 3, the authors introduced the operator Djm 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 whichf0 h0 fz0−1 0. For more details about the operator

Djm, see4.

Now, we defineDjmforfhggiven by1.3as

Djmfz Djmhz −1jDjmgz, j, m∈N0N∪ {0};z∈U\ {0}, 1.4

where

Djmhz −1 j

z

∞

n1

nj_C_{m, n}_a

nzn,

Djmgz ∞

n1

nj_C_{m, n}_b

nzn,

Cm, n

nm−1

m

nm−1!

m!n−1! .

1.5

A functionf ∈ MHis said to be in the subclassMS∗Hof meromorphically harmonic starlike functions inU\ {0}if it satisfies the condition

Re

−zhz−zgz

hz gz

>0, z∈U\ {0}. 1.6

Note that the class of harmonic meromorphic starlike functions has been studied by Jahangiri and Silverman5, and Jahangiri6.

Now, we have the following definition.

Definition 1.1. For j, m ∈ N0, 0 ≤ α < 1 andk ≥ 1, letMHSskj, m, αdenote the class of meromorphic harmonic functionsfof the form1.3such that

Re

−D

j1 m fz

Djmfkz

>0, z∈U\ {0}, 1.7

where

Djmfkz Djmhk −1jDjmgk j, m∈N0, k≥1, 1.8

hkz −1

j

z

∞

n1

anΦnzn, gkz

∞

n1

Φnzn, 1.9

Φn 1_k k−1

ν0

εn−1ν_, _k_≥_1;_ε_exp2πi

k

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For more details about harmonic functions with respect to k-symmetric points, see papers7,8given by the authors.

Also, note thatMHSs2j,0, α⊂MHS∗_Sn, αwas introduced by Bostancıand ¨Ozt ¨urk 9.

Finally, letMHSskj, m, αdenote the subclass ofMHSskj, m, αconsist of harmonic functionsfjhjgjsuch thathjandgjare of the form

hjz −1

j

z

∞

n1

|an|zn, gjz −1j ∞

n1

|bn|zn. 1.11

Also, letfkjhkjgkj wherehkjandgkjare of the form

hkjz

−1j

z

∞

n1

Φn|an|zn, gkjz −1j

∞

n1

Φn|bn|zn, 1.12

whereΦnis given by1.10.

In this paper, we will give a suﬃcient condition for functionsf hg,wherehand

g given by1.3to be in the classMHSskj, m, α. Indeed, it is shown that this coeﬃcient condition is also necessary for functions to be in the classMHSskj, m, α. Also, we obtain distortion bounds and characterize the extreme points for functions in MHSskj, m, α. Convolution and closure theorems are also obtained.

2. Coefficient bounds

First, we prove a suﬃcient coeﬃcient bound.

Theorem 2.1. Letf hg be of the form1.3andfk hkgk wherehk andgk are given by 1.9. If

∞

n1

n−1k1α|a_n−1k1| n−1k1−α|bn−1k1|Ωjmn, k

∞ n2 n /lk1

nj1_C_{m, n}_|_a

n||bn|≤1−α,

2.1

wherej, m∈N0, 0≤α <1, k≥1andΩjmn, k n−1k1jCm, nk1, thenfis harmonic univalent, sense preserving inU\ {0}andf∈ MHSskj, m, α.

Proof. For 0<|z1| ≤ |z2|<1, we have f z1

−f z2

≥h z1

−h z2−g z1

−g z2

≥ z1−z2

z1z2

−z1−z2 ∞

n1

anbnzn−1₁ · · ·zn−1₂

> z1−z2

z1z2

1−z22 ∞

n1

n anbn

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> z1−z2

z1z2

1−z22 _∞

n1

n anbn

∞

n1

n−1k1 an−1k1bn−1k1

> z1−z2

z1z2

1−

∞

n1

n−1k1αan−1k1−n−1k1−αbn−1k1

Ωjmn, k

− ∞

n2 n /lk1

nj1_C_{m, n}_a

nbn.

2.2

This last expression is nonnegative by2.1, and sofis univalent inU\ {0}. To show thatf is sense preserving inU\ {0}, we need to show that|hz| ≥ |gz|inU\ {0}. We have

|h_z_{| ≥} 1

|z|2 − ∞

n1

n|an||z|n−1

1

r2 −

∞

n1

n|an|rn−1>1− ∞

n1

n|an|

≥1− ∞

n1

n−1k1α|a_n−1k1|Ωjmn, k− ∞

n2 n /lk1

nj1_C_{m, n}_|_a n|

≥∞

n1

n−1k1−α|bn−1k1|Ωjmn, k ∞

n2 n /lk1

nj1_C_{m, n}_|_b n|

≥∞

n1

2n|b2n| ∞

n1

2n−1|b2n−1|

>∞

n1

n|bn|rn−1 ∞

n1

n|bn||z|n−1≥ |gz|.

2.3

Now, we will show thatf ∈ MHSskj, m, α. According to1.4and1.7, for 0≤α <1, we have

Re

−D

j1 m fz

Djmfkz

Re ⎧ ⎨ ⎩−

Djm1hz−−1jDjm1gz

Djmhkz −1jDjmgkz ⎫ ⎬

⎭≥α. 2.4

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

1−α−D j1 m fz

Djmfkz

≥1αD j1 m fz

Djmfkz

, 2.5

which is equivalent to _Dj1

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SubstitutingDjmfz, Djm1fz,andDjmfkzin2.6yields

Dj1

m hz−−1jDmj1gz−1−α

Djmhkz −1jDjmgkz

−Dj1

m hz−−1jDmj1gz 1α

Djmhkz −1jDjmgkz

−_z1j −∞

n1

nj1_C_{m, n}_a

nzn −1j ∞

n1

nj1_C_{m, n}_b nzn

1−α

−1j

z

∞

n1

nj_C_{m, n}_Φ

nanzn −1j ∞

n1

nj_C_{m, n}_Φ

nbnzn

−−1j

z −

∞

n1

nj1_C_{m, n}_a

nzn −1j ∞

n1

nj1_C_{m, n}_b nzn

−1α

−1j

z

∞

n1

nj_C_{m, n}_Φ

nanzn −1j ∞

n1

nj_C_{m, n}_Φ

nbnzn

2−α_z−1j−∞

n1

nj_C_{m, n}_n₋₁₋_α_Φ

nanzn−1j ∞

n1

nj_C_{m, n}_n₁₋_α_Φ

nbnzn

−α−1j

z −

∞

n1

nj_C_{m, n}_n₁_α_Φ

nanzn −1j ∞

n1

nj_C_{m, n}_n₋₁_α_Φ

nbnzn

≥ 2_|−_z_|α−∞

n1

nj_C_{m, n}_n₋₁₋_α_Φ

n|an||zn| − ∞

n1

nj_C_{m, n}_n₁₋_α_Φ

n|bn||zn|

−_|α_z_|−∞

n1

nj_C_{m, n}_n₁_α_Φ

n|an||zn| − ∞

n1

nj_C_{m, n}_n₋₁_α_Φ

n|bn||zn|

21_|_z−_|α

1− ∞

n1

nj_C_{m, n}_n_α_Φ

n 1−α |an|z

n1₋∞ n1

nj_C_{m, n}_n₋_α_Φ

n 1−α |bn|z

n1

≥21−α

1− ∞

n1

n 1−α |an| −

∞

n1

n 1−α |bn|

.

2.7

From the definition ofΦn,we know that

Φn ⎧ ⎨ ⎩

1, nlk1,

0, n /lk1,

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Substituting2.8in2.7, then2.7is equivalent to _Dj1

m fz−1−αDjmfkz−Djm1fz 1αDjmfkz

≥21−α

1− ∞

n1

nk1jCm, nk1nk1α

1−α |ank1|

−∞

n1

nk1jCm, nk1nk1−α

1−α |bnk1| − ∞

n2 n /lk1

nj_C_{m, n}

1−α |an|

− ∞

n2 n /lk1

nj_C_{m, n}

1−α |bn| − 1α

1−α|a1| − |b1|

21−α

1− ∞

n1

_n₋₁_k 1α

1−α |an−1k1| −

n−1k1−α

1−α |bn−1k1|

Ωjmn, k

− ∞

n2 n /lk1

nj1_C_{m, n}

1−α |an||bn|

≥0, by2.6.

2.9 Thus, this completes the proof of the theorem.

We next show that condition2.1is also necessary for functions inMHSskj, m, α.

Theorem 2.2. Letfj hjgj, wherehj andgjare given by1.11, andfkj hkjgkj wherehkj

andgkj are given by1.12. Then,fj ∈ MHS

k

s j, m, α, if and only if the inequality2.1holds for

the coeﬃcient offjhjgjandfkj hkjgkj.

Proof. In view ofTheorem 2.1, we need only to show thatfj/∈MHSskj, m, αif condition 2.1does not hold. We note that forfj ∈ MHSskj, m, α, then by1.7the condition2.4 must be satisfied for all values ofzinU\ {0}. Substituting forhj, gj, hkj,andgkj given by

1.11and1.12, respectively, in2.4and choosing 0< z r < 1, we are required to have Re{Ψz/Υz} ≥0, where

Ψz −Dj1

m hjz −1nDjm1gjz−αDjmhkjz−α−1

j_Dj mgkjz

1−_zα−∞

n1

n|an|zn ∞

n1

n|bn|zn,

Υz Djmhkjz −1

j_Dj mgkjz

_z1 ∞ n1

nj_C_{m, n}_Φ

n|an|zn ∞

n1

nj_C_{m, n}_Φ

n|bn|zn.

2.10

Then, the required condition Re{Ψz/Υz} ≥0 is equivalent to 1−α/z−∞

n1njCm, nnαΦn|an|rn _∞

n1njCm, nn−αΦn|bn|rn 1/z∞_n₁njCm, nΦn|an|rn∞n1njCm, nΦn|bn|rn

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By using2.8, and if condition2.1does not hold, then the numerator of2.11is negative forrsuﬃciently close to 1. Thus, there exists az0 r0in0,1for which the quotient in2.11 is negative. This contradicts the required condition forfj ∈ MHSskj, m, αand so the proof is complete.

3. Distortion bounds and extreme points

In this section, we will obtain distortion bounds for functionsfj ∈ MHSskj, m, αand also provide extreme points for the classMHSskj, m, α.

Theorem 3.1. Iffjhjgj∈ MHSskj, m, αand0<|z|r <1, then

1

r −

1−α

2jm12−αr≤ |fjz| ≤ 1

r

1−α

2jm12−αr. 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. Letfj hj gj ∈

MHSskj, m, α. Taking the absolute value off, we obtain

|fj| −1

j

z

∞

n1

anzn −1n

∞

n1

bnzn

≥ 1_r −∞

n1

|an||bn|rn

≥ 1_r −∞

n1

|an||bn|r

≥ 1

r −

1−α 2jm12−αΦ2

∞

n1

2jm12−αΦ2

1−α |an||bn|r

≥ 1

r −

1−α 2jm12−α

∞

n1

_n_j_C_{m, n}_n_α_Φ n 1−α |an|

n 1−α |bn|

r

≥ 1_r − 1−α

2jm12−αr, by2.7.

3.2

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

2.1is satisfied.

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

Theorem 3.1.

Corollary 3.2. Iffj∈ MHSskj, m, α, then

fjU\ {0}⊂

w:|w|< 2

j_m₁₂₋_α₋₁₋_α 2jm12−α

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Next, we determine the extreme points of closed convex hulls of MHSskj, m, α denoted by clcoMHSskj, m, α.

Theorem 3.3. Letfj hjgjwherehjandgjare given by1.11. Then,fj ∈ MHSskj, m, αif and only if

fj,nz ∞

n0

xnhjnz yngjnz, 3.4

where hj,0 gj,0z −1j/z, hj,nz −1j/z 1−α/njCm, nnαΦnzn n

1,2,3, . . ., gj,nz −1j/z−1j1−α/njCm, nn−αΦnzk n1,2,3, . . ., ∞n0xn

yn 1, xn ≥0, yn≥0. In particular, the extreme points ofMHSskj, m, αare{hj,n}and{gj,n}.

Proof. For functionsfjhjgj, wherehjandgjare given by1.11, we have

fj,nz ∞

n0

xnhj,nz yngj,nz

∞ n0

xnyn−1 j

z

∞

n1

1−α

nj_C_{m, n}_n_α_Φ_nxnzn

−1j∞ n1

1−α

nj_C_{m, n}_n₋_α_Φ_nynzk.

3.5

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

∞

n1

1−α

nj_C_{m, n}_n_α_Φ_n

n 1−α xn

∞ n1

1−α

nj_C_{m, n}_n₋_α_Φ_n

n 1−α yn

∞ n0

xnyn−x0y0 1−x0y0≤1.

3.6

Conversely, suppose thatfj∈clcoMHSskj, m, α. Forn1,2,3, . . ., set

xn n

j_C_{m, n}_n_α_Φ_n

1−α |an| 0≤xn≤1,

yn n

j_C_{m, n}_n₋_α_Φ n

1−α |bn| 0≤yn≤1,

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x01−∞n1xnyn. Therefore,fcan be written as

fj,nz −1

j

z

∞

n1

|an|zn −1j ∞

n1

|bn|zn

−1j

z

∞

n1

1−αxn

nj_C_{m, n}_n_α_Φ_nzn −1j

∞

n1

1−αyn

nj_C_{m, n}_n₋_α_Φ_nzn

−_z1j ∞ n1

hj,nz−−1

j z xn ∞

n1

gj,nz−−1

j

z

yn

∞ n1

hj,nzxn

∞

n1

gj,nzyn −1

j z 1− ∞

n1

xn−

∞

n1

yn

∞ n0

hj,nzxngj,nzyn, as required.

3.8

4. Convolution and convex combination

In this section, we show that the classMHSskj, m, αis invariant under convolution and convex combination of its member.

For harmonic functionsfjz −1j/z∞n1|an|zn −1j _∞

n1|bn|zn andFjz −1j_/z∞

n1|An|zn −1j _∞

n1|Bn|z

n_{, the convolution of}_f

jandFjis given by

fj∗Fjz fjz∗Fjz −1 j

z

∞

n1

|an||An|zn −1j ∞

n1

|bn||Bn|zn. 4.1

Theorem 4.1. For0 ≤ β ≤ α < 1, letfj ∈ MHSskj, m, αand Fj ∈ MHSskj, m, β. Then,

fj∗Fj∈ MHSskj, m, α⊂ MHSskj, m, β.

Proof. We wish to show that the coeﬃcients offj∗Fj satisfy the required condition given in

Theorem 2.2. ForFj ∈ MHSskj, m, β, we note that |An| ≤ 1 and |Bn| ≤ 1. Now, for the convolution functionfj∗Fj, we obtain

∞

n1

nj_C_{m, n}_n_β_Φ

n

1−β |an||An| ∞

n1

nj_C_{m, n}_n₋_β_Φ

n 1−β |bn||Bn|

≤∞

n1

nj_C_{m, n}_n_β_Φ_n

1−β |an| ∞

n1

nj_C_{m, n}_n₋_β_Φ_n

1−β |bn|

≤∞

n1

n 1−α |an|

∞

n1

n

1−α |bn| ≤1,

4.2

since 0 ≤ β ≤ α < 1 and fj ∈ MHSskj, m, α. Therefore fj∗Fj ∈ MHSskj, m, α ⊂

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We now examine the convex combination ofMHSskj, m, α. Let the functionsfj,tbe defined, fort1,2, . . . , ρ,by

fj,tz −1

j

z

∞

n1

|an,t|zn −1j ∞

n1

|bn,t|zn. 4.3

Theorem 4.2. Let the functionsfj,tdefined by4.3be in the classMHSskj, m, αfor everyt

1,2, . . . , ρ. Then, the functionsξtzdefined by

ξtz ρ

t1

ctfjnz, 0≤ct≤1, 4.4

are also in the classMHSskj, m, α,whereρ_t₁ct1.

Proof. According to the definition ofξt, we can write

ξtz −1

j

z

∞

n1 _ρ

t1

ctan,t

zn ₋₁j∞

n1 _ρ

t1

ctbn,t

zn. 4.5

Further, sincefj,tzare inMHSskj, m, αfor everyt1,2, . . . , ρ. Then by2.7, we have

∞

n1

nαΦn _ρ

t1

ct|an,t|

n−αΦn

_ρ

t1

ct|bn,t|

nj_C_{m, n}

ρ

t1

ct _∞

n1

nαΦn|an,t| n−αΦn|bn,t|njCm, n

≤

ρ

t1

ct1−α≤1−α.

4.6

Hence, the theorem follows.

Corollary 4.3. The classMHSskj, m, αis close under convex linear combination.

Proof. Let the functionsfj,tz t1,2defined by4.3be in the classMHSskj, m, α. Then, the functionψzdefined by

ψz μfj,1z 1−μfj,2z, 0≤μ≤1, 4.7

is in the classMHSskj, m, α. Also, by takingρ2, ξ1μ,andξ2 1−μinTheorem 4.2, we have the corollary.

Acknowledgment

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