Subclass of Multivalent Harmonic Functions with Missing Coefficients

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Volume 2012, Article ID 915625,10pages doi:10.1155/2012/915625

Research Article

Subclass of Multivalent Harmonic Functions with

Missing Coefficients

R. M. El-Ashwah

Department of Mathematics, Faculty of Science (Damietta Branch), Mansoura University, New Damietta 34517, Egypt

Correspondence should be addressed to R. M. El-Ashwah,r [email protected]

Received 20 March 2012; Accepted 8 July 2012

Academic Editor: Attila Gil´anyi

Copyrightq2012 R. M. El-Ashwah. 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.

We have studied subclass of multivalent harmonic functions with missing coeﬃcients in the open unit disc and obtained the basic properties such as coeﬃcient characterization and distortion theorem, extreme points, and convolution.

1. Introduction

A continuous function f u iv is a complex-valued harmonic function in a simply connected complex domain D ⊂ C if bothuand vare real harmonic in D. It was shown by Clunie and Sheil-Small1that such harmonic function can be represented byf hg, wherehandgare analytic inD. Also, a necessary and suﬃcient condition forfto be locally univalent and sense preserving inDis that|hz|>|gz|see also,2–4.

Denote by Hthe family of functions f hg, which are harmonic univalent and sense-preserving in the open-unit discU{z∈C:|z|<1}with normalizationf0 h0

f

z0−10.

Form ≥ 1, 0 ≤ β < 1,and γ ≥ 0,let Rm, β, γ denote the class of all multivalent harmonic functionsf hg with missing coeﬃcients that are sense-preserving inU, and

h, gare of the form

hz zm ∞

nm1

an1zn1, gz

∞

nm

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and satisfying the following condition:

Re1γeiφzf _z

z_f_z−γmeiφ

≥mβ m≥1; 0≤β <1; γ≥0; φ real, 1.2

where

z ∂

∂θ

zreiθ_, _f_z ∂

∂θf

reiθ_. _1.3

We note that:

iRm, β,1 Rm, βwitham1; bm/0see Jahangiri et al.5;

iiR1, β, γ JHα, β, γ see Kharinar and More6;

iiiR1, β,1 GHβ witha2; b1/0see Rosy et al.7and Ahuja and Jahangiri2.

Also, the subclass denoted byTm, β, γconsists of harmonic functionsf hg, so thathandgare of the form

hz zm₋ ∞ nm1

an1zn1, gz

∞

nm

bn1zn1 an1; bn1≥0; m≥1; z∈U. 1.4

We note that:

iTm, β,1 Tm, βwitham1;bm/0 see Jahangiri et al.5;

iiT1, β, γ JHα, β, γ see Kharinar and More6;

iiiT1, β,1 G_Hβwitha2;b1/0 see Rosy et al.7and Ahuja and Jahangiri2.

From Ahuja and Jahangiri2with slight modification and among other things proved, iffhg is of the form1.1and satisfies the coeﬃcient condition

∞

nm−1

n1−mβ

m1−β |an1|

n1 mβ

m1−β |bn1|

≤2 am1;am1bm0, 1.5

then the harmonic function f is sense-preserving, harmonic multivalent with missing coeﬃcients and starlike of orderβ0 ≤ β < 1 inU. They also proved that the condition

1.5is also necessary for the starlikeness of functionfhg of the form1.4.

In this paper, we obtain suﬃcient coeﬃcient bounds for functions in the class

Rm, β, γ. These suﬃcient coeﬃcient conditions are shown to be also necessary for functions in the class Tm, β, γ. Basic properties such as distortion theorem, extreme points, and convolution for the classTm, β, γare also obtained.

2. Coefficient Characterization and Distortion Theorem

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Theorem 2.1. Letf hgbe such thathandgare given by1.1. Furthermore, let

∞

nm−1

1γn1−mγβ

m1−β |an1|

1γn1 mγβ

m1−β |bn1|

≤2, 2.1

where am 1andam1 bm 0. Thenf is sense-preserving, harmonic multivalent in Uand

f∈Rm, β, γ.

Proof. To provef ∈Rm, β, γ, by definition, we only need to show that the condition2.1

holds forf. Substitutinghg forfin1.2, it suﬃces to show that

Re

⎧ ⎨ ⎩

1γeiθzhz−zgz−mβγeiθhz gz

hz gz

⎫ ⎬

⎭≥0, 2.2

wherehz ∂/∂zhzandgz ∂/∂zgz. Substituting forh, g, h, andg in2.2, and dividing bym1−βzm, we obtain ReAz/Bz≥0, where

Az 1

∞

nm1

n11γeiθ−mβγeiθ

m1−β an1z

n−m1

−

_z

z

m ∞

nm1

n11γe−iθmβγe−iθ

m1−β bn1z

n−m1_,

Bz 1

∞

nm1

an1zn−m1

_z

z

m ∞

nm1

bn1zn−m1.

2.3

Using the fact that Rew≥0 if and only if|1w|>|1−w|inU, it suﬃces to show that |Az

Bz| − |Az−Bz| ≥0. Substituting forAzandBzgives

|Az Bz| − |Az−Bz|

2 ∞

nm1

n11γeiθ−m−12βγeiθ

m1−β an1z

n−m1

−

_z

z

m∞

nm

n11γe−iθm−12βγe−iθ

m1−β bn1z

n−m1

− ∞

nm1

n11γeiθ−m1γeiθ

m1−β an1z

n−m1

−

_z

z

∞

nm

n11γe−iθm1γe−iθ

m1−β bn1z

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≥2− ∞

nm1

n11γ−m2βγ−1

m1−β |an1| |z|

n−m1

−∞

nm

n11γm2βγ−1

m1−β |bn1| |z|

n−m1

− ∞

nm1

n11γ−m1γ

m1−β |an1| |z|

n−m1

−∞

nm

n11γm1γ

m1−β |bn1| |z|

n−m1

≥2

1− ∞

nm1

n11γ−mβγ

m1−β |an1|

−∞

nm

n11γmβγ

m1−β |bn1|

≥0 by2.1

2.4

The harmonic functions

fz zm ∞

nm1

m1−β

n11γ−mβγxnz

n1

∞

nm

m1−β

n11γmβγynz

n1_,

2.5

where∞_n_m₁|xn|n∞m|yn|1, show that the coeﬃcient boundary given by2.1is sharp.

The functions of the form2.5are in the classRm, β, γbecause

∞

nm1

1γn1−mβγ

m1−β |an1| ∞

nm

1γn1 mβγ

m1−β |bn1|

∞

nm1

|xn|

∞

nm1

_y_n₁_. 2.6

This completes the proof ofTheorem 2.1.

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Theorem 2.2. Letfhgbe such thathandgare given by1.4. Thenf ∈Tm, β, γif and only if

∞

nm−1

1γn1−mγβ

m1−β an1

1γn1 mγβ

m1−β bn1

≤2. 2.7

Proof. SinceRm, β, γ⊂Tm, β, γ, we only need to prove the “only if” part of the theorem.

To this end, for functions f of the form 1.4, we notice that the condition Re{1

γeiθ_zf_z_/_z_f_z₋_γmeiθ_{} ≥}_mβ_{is equivalent to}

Re

⎧ ⎨ ⎩

1γeiθzhz−zgz−mβγeiθhz gz

hz gz

⎫ ⎬

⎭>0, 2.8

which implies that

Re

m1γeiθ−mβγeiθzm−∞_n_m₁1γeiθn1−mβγeiθan1zn1

zm₋∞

nm1an1zn1∞nmbn1zn1

−

_∞

nm1γeiθn1 mβγeiθbn1zn1

zm₋∞

nm1an1zn1∞nmbn1zn1

Re

m1−β−∞_n_m₁1γeiθn1−mβγeiθan1zn−m1

1−∞_n_m₁an1zn−m1∞nmbn1zn−m1

−

_∞

nm1γeiθn1 mβγeiθbn1zn−m1

1−∞_n_m₁an1zn−m1∞nmbn1zn−m1

>0.

2.9

Since Reeiθ≤ |eiθ|1, the required condition is that2.9is equivalent to

1−∞_n_m₁1γn1−mβγ/m1−βan1rn−m1

1−∞_n_m₁an1rn−m1∞nmbn1rn−m1

−

_∞

nm

1γn1 mβγ/m1−βbn1rn−m1

1−∞_n_m₁an1rn−m1∞nmbn1rn−m1

≥0.

2.10

If the condition 2.7 does not hold, then the numerator in 2.10 is negative for z r suﬃciently close to 1. Hence there exists z0 r0 in 0,1 for which the quotient in 2.10

is negative. This contradicts the required condition forf ∈ Tm, β, γ, and so the proof of Theorem 2.2is completed.

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Proof. The proof follows from1.5, by putting2.7in the form

∞

nm−1

n1−mγβ/1γ

m1−γβ/1γ an1

n1 mγβ/1γ

m1−γβ/1γ bn1

≤2. 2.11

Theorem 2.4. Letf ∈Tm, β, γ. Then for|z|r <1, we have

fz≤1bm1rrm

m1−β

m1−β21γ−

m12γβ1γ

m1−β21γ bm1

rm2_,

fmz≥1−bm1rrm−

m1−β

m1−β21γ−

m12γβ1γ

m1−β21γ bm1

rm2_.

2.12

Proof. We prove the left-hand-side inequality for |f|. The proof for the right-hand-side

inequality can be done by using similar arguments. Letf∈Tm, β, γ, then we have

_f_z

zm−

∞

nm1

an1zn1

∞

nm

bn1zn1

≥rm₋_b

m1rm1−

∞

nm1

an1bn1rm2

≥rm₋_b m1rm1

− m

1−β

1γm2−mγβ ∞

nm1

1γm2−mγβ

m1−β an1bn1r

n1

≥rm₋_b m1rm1

− m

1−β

1γm2−mγβ ∞

nm1

1γn1−mγβ

m1−β an1

1γn1 mγβ

m1−β bn1

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≥1−bm1rrm

− m

1−β

1γm2−mγβ

1−

1γm1 mγβ

m1−β bm1

rm2

≥1−bm1rrm

−

m1−β

m1−β21γ −

m12γβ1γ

m1−β21γ bm1

rm2_.

2.13

This completes the proof ofTheorem 2.4.

The following covering result follows from the left-side inequality inTheorem 2.4.

Corollary 2.5. Letf ∈Tm, β, γ,then the set

w:|w|< 2

1γ

m1−β21γ−

1γ−2mγβ

m1−β21γbm1

2.14

is included infU.

3. Extreme Points

Our next theorem is on the extreme points of convex hulls of the classTm, β, γ, denoted by clcoTm, β, γ.

Theorem 3.1. Letfhgbe such thathandgare given by1.4. Thenf∈clcoTm, β, γif and

only iffcan be expressed as

fz ∞

nm

Xn1hn1z Yn1gn1z, 3.1

where

hmz zm,

hn1z zm− m

1−β

1γn1−mγβz

n1 _n_m₁_{, m}₂_{, ...}_,

gn1z zm m

1−β

1γn1 mγβz

n1 _n_{m, m}₁_{, m}₂_{, ...}_,

Xn1≥0, Yn1≥0,

∞

nm

Xn1Yn1 1.

3.2

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Proof. For functionsfzof the form3.1, we have

fz ∞

nm

Xn1Yn1zm−

∞

nm

m1−β

1γn1−mγβXn1z

n1

∞

nm

m1−β

1γn1 mγβYn1z

n1_.

3.3

Then

∞

nm1

1γn1−mγβ

m1−β

1γn1−mγβ

Xn1

∞

nm

1γn1 mγβ

m1−β

1γn1 mγβ

Yn1

∞

nm1

Xn1

∞

nm

Yn1 1−Xm≤1,

3.4

and sofz∈clcoTm, β, γ. Conversely, suppose thatfz∈clcoTm, β, γ. Set

Xn1

1γn1−mγβ

m1−β an1 nm1, ...,

Yn1

1γn1 mγβ

m1−β bn1 nm, m1, ...,

3.5

then note that byTheorem 2.2, 0≤Xn1≤1 nm1, ...and 0≤Yn1≤1 nm, m1, ....

Consequently, we obtain

fz ∞

nm

Xn1hn1z Yn1gn1z. 3.6

Using Theorem 2.2 it is easily seen that the class Tm, β, γ is convex and closed, and so clcoTm, β, γ Tm, β, γ.

4. Convolution Result

For harmonic functions of the form

fz zm₋ ∞ nm1

an1zn1

∞

nm

bn1zn1, 4.1

Gz zm₋ ∞ nm1

An1zn1

∞

nm

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we define the convolution of two harmonic functionsfandGas

f∗Gz fz∗Gz

zm₋ ∞ nm1

an1An1zn1

∞

nm

bn1Bn1zn1.

4.3

Using this definition, we show that the classTm, β, γis closed under convolution.

Theorem 4.1. For0 ≤ β < 1, letfz ∈Tm, β, γand Gz ∈ Tm, β, γ. Thenfz∗Gz ∈

Tm, β, γ.

Proof. Let the functionsfzdefined by4.1be in the class Tm, β, γ, and let the functions

Gzdefined by4.2be in the classTm, β, γ. Obviously, the coeﬃcients off andGmust satisfy a condition similar to the inequality2.7. So for the coeﬃcients off∗G we can write

∞

nm−1

1γn1−mγβ

m1−β an1An1

1γn1 mγβ

m1−β bn1Bn1

≤ ∞

nm−1

1γn1−mγβ

m1−β an1

1γn1 mγβ

m1−β bn1

,

4.4

where the right hand side of this inequality is bounded by 2 becausef ∈ Tm, β, γ. Then,

fz∗Gz∈Tm, β, γ.

Finally, we show thatTm, β, γis closed under convex combinations of its members.

Theorem 4.2. The classTm, β, γ is closed under convex combination.

Proof. Fori1,2,3, .... letfi∈Tm, β, γ,where the functionsfiare given by

fiz zm−

∞

nm1

an1,izn1

∞

nm

bn1,izn1 an1,i;bn1,i≥0;m≥1. 4.5

For∞_i₁ti 1; 0≤ti≤1,the convex combination offimay be written as

∞

i1

tifiz zm−

∞

nm1

_∞

i1

tian1,i

zn1∞

nm

_∞

i1

tibn1,i

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Then by2.7, we have

∞

nm−1

1γn1−mγβ

m1−β

∞

i1

tian1,i

1γn1 mγβ

m1−β

∞

i1

tibn1,i

∞

i1

ti

_∞

nm−1

1γn1−mγβ

m1−β an1,i

1γn1 mγβ

m1−β bn1,i

≤2 ∞

i1

ti2.

4.7

This is the condition required by2.7, and so∞i1tifiz ∈ Tm, β, γ. This completes the

proof ofTheorem 4.2.

Remark 4.3. Our results form1 correct the results obtained by Kharinar and More6.

Acknowledgment

The author thanks the referees for their valuable suggestions which led to the improvement of this study.

References

1 J. Clunie and T. Sheil-Small, “Harmonic univalent functions,”Annales Academiae Scientiarum Fennicae A, vol. 9, pp. 3–25, 1984.

2 O. P. Ahuja and J. M. Jahangiri, “Multivalent harmonic starlike functions,”Annales Universitatis Mariae Curie-Skłodowska A, vol. 55, pp. 1–13, 2001.

3 J. M. Jahangiri, “Coeﬃcient bounds and univalence criteria for harmonic functions with negative coeﬃcients,”Annales Universitatis Mariae Curie-Skłodowska A, vol. 52, no. 2, pp. 57–66, 1998.

4 H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,”New Zealand Journal of Mathematics, vol. 28, no. 2, pp. 275–284, 1999.

5 J. M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, “On starlikeness of certain multivalent harmonic functions,”Journal of Natural Geometry, vol. 24, no. 1-2, pp. 1–10, 2003.

6 S. M. Kharinar and M. More, “Certain class of harmonic starlike functions with some missing coeﬃcients,”Acta Mathematica, vol. 24, no. 3, pp. 323–332, 2008.

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Subclass of Multivalent Harmonic Functions with Missing Coefficients

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