Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product

(1)

Volume 2009, Article ID 759251,12pages doi:10.1155/2009/759251

Research Article

Certain Classes of Harmonic Multivalent Functions

Based on Hadamard Product

Om P. Ahuja,

1

_{H. ¨}

_{Ozlem G ¨uney,}

2

_{and F. M ¨uge Sakar}

2

1_{Department of Mathematical Sciences, Kent State University, 14111 Claridon-Troy Road,} Burton, OH 44021, USA

2_{Department of Mathematics, Faculty of Science and Arts, Dicle University,} 21280 Diyarbakır, Turkey

Correspondence should be addressed to H. ¨Ozlem G ¨uney,[email protected]

Received 25 February 2009; Accepted 12 September 2009

Recommended by Yong Zhou

We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product.

Copyrightq2009 Om P. Ahuja et al. 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 complex domain

Cif bothuandvare real harmonic inC. In any simply connected domainD⊆C, we can write

f hg, wherehandgare analytic inD. We callhthe analytic part andg the co-analytic part off. Note thatfhgreduces tohif the coanalytic partgis zero.

For p ≥ 1, denote byHp the set of all multivalent harmonic functionsf hg defined in the open unit discU, wherehandgdefined by

hz zp ∞ npt

anzn, gz

∞

npt−1

bnzn, bpt−1<1, t∈N:{1,2, . . .} 1.1

are analytic functions inU.

LetFbe a fixed multivalent harmonic function given by

Fz Hz Gz zp ∞ npt

|An|zn

∞

npt−1

(2)

A functionf ∈Hpis said to be in the classHFp, t, α, kif

Re

zf∗Fz z_f_∗_F_z

≥kz

f∗Fz z_f_∗_F_z−p

pα , 1.3

where f∗F is a harmonic convolution off and F. Note thatz ∂/∂θr eiθ, fz

∂/∂θfr eiθ_{. Using the fact}

Re w > kw−ppα⇐⇒Re1keiθ w−kpeiθ ≥pα, 1.4

it follows thatf∈HFp, t, α, kif and only if

Re

1keiθzf∗Fz

z_f_∗_F_z −kpe

iθ

≥pα, 0≤α <1. 1.5

A function f in HFp, t, α, k is called k-uniformly multivalent harmonic starlike

function associated with a fixed multivalent harmonic function F. The set HFp, t, α, k is

a comprehensive family that contains several previously studied subclasses of Hp; for example, if we let

Fz Iz:

_zp

1−z

_zp

1−z

zp ∞ np1

zn∞ np

zn_, _1.6

then

HIp, α,0≡S∗H

p, α:

f∈Hp: Re

_{z f}_z z_f_z

≥pα

, 1.7

see1,2;

HIp,1,0,0≡S∗H

p,0≡S∗_Hp, 1.8

see3;

HI1,1, α,0≡SH∗ 1, α≡S∗Hα, 1.9

see4;

(3)

see5,6;

HI1,1, α, k≡GHk, α:

f∈H1: Re

1keiθzfz

z_fz −keiθ

≥α

, 1.11

see7;

HIp,1, α,1≡MHp, α:

f ∈Hp: Re1eiθ zf

_z

z_fz−peiθ

≥pα

, 1.12

see8.

Finally, denote byTHpthe subclass of functionsfz hz gzinHpwhere

hz zp₋ ∞ npt

|an|zn, gz

∞

npt−1

|bn|zn. 1.13

LetH_Fp, t, α, k:THp∩HFp, t, α, k.

In this paper, we investigate coeﬃcient conditions, extreme points, and distortion bounds for functions in the familyH_Fp, t, α, k. We observe that the results so obtained for this main family can be viewed as extensions and generalizations for various subclasses of

HpandH1.

2. Main Results

Theorem 2.1. Letf hgbe such thathandgare given by1.1. Thenf ∈HFp, t, α, kif the

inequality

∞

npt

n1k−pkα

p1−α 1−p1−α−1

|anAn|

∞

npt−1

n1k pkα

p1−α 1−p1−α−1

|bnBn| ≤ 1

2

2.1

is satisfied for somek k≥0, p p≥1, α0≤α <1,andtt≥1.

Proof. In view of1.5, we need to prove that Rew>0, where

w

keiθ₁_zh_∗_H_z₋_z_g_∗_G_z₋_p_keiθ_α_h_∗_Hz_g_∗_G_z

h∗Hz g∗Gz : Az Bz.

2.2

Using the fact that Rew>0⇔ |1w| ≥ |1−w|, it suﬃcies to show that

(4)

Therefore, we obtain

|Az Bz| − |Az−Bz|

≥p1−α 1−p1−α−1|z|p−

∞

npt

n1k−pkα 1|anAn||z|n

− ∞

npt−1

n1k pkα−1|bnBn||z|n−

∞

npt

n1k−pkα−1|anAn||z|n

− ∞

npt−1

n1k pkα 1|bnBn||z|n

p1−α 1−p1−α−1|z|p

− ∞

npt

2n1k−pkα|anAn||z|n−

∞

npt−1

2n1k pkα|bnBn||z|n

p1−α 1−p1−α−1|z|p

× ⎧ ⎨ ⎩1−

∞

npt

2n1k−pkα

p1−α 1−p1−α−1|anAn|

− ∞

npt−1

2n1k pkα

p1−α 1−p1−α−1|bnBn|

⎫ ⎬ ⎭≥0.

2.4

By hypothesis, last expression is nonnegative. Thus the proof is complete.

The coeficient bounds2.1is sharp for the function

fz zp ∞ npt

p1−α 1−p1−α−1 2n1k−pkα Xnz

n

∞

npt−1

p1−α 1−p1−α−1 2n1k pkαBn Ynz

n_,

2.5

where∞_n_p_t|Xn|∞npt−1|Yn|1.

Corollary 2.2. Forp≥1/1−α,0≤α <1, if the inequality

∞

npt

n1k−pkα|anAn|

∞

npt−1

n1k pkα|bnBn| ≤1 2.6

(5)

Corollary 2.3. For0≤α <1and1≤p≤1/1−α, if the inequality

∞

npt

∞

npt−1

n1k pkα|bnBn| ≤p1−α 2.7

holds, thenf ∈HFp, t, α, k.

Theorem 2.4. Letfhgbe such thathandgare given by1.13. Also, suppose thatk≥0and

0≤α <1. Then

ifor1≤p≤1/1−α,f∈H_F p, t, α, kif and only if

∞

npt

∞

npt−1

n1k pkα|bnBn| ≤p1−α; 2.8

iiforp1−α≥1,f∈H_F p, t, α, kif and only if

∞

npt

∞

npt−1

n1k pkα|bnBn| ≤1. 2.9

Proof. According to Corollaries2.2and2.3, we must show that if the condition2.9does not hold, thenf /∈H_Fp, t, α, k,that is, we must have

Re

⎛ ⎜ ⎜ ⎝

1keiθzh∗Hz−zg∗Gz

−pkeiθ_α_h_∗_Hz_g_∗_G_z

h∗Hz g∗Gz

⎞ ⎟ ⎟ ⎠≥0.

2.10

Choosing the values ofzron positive real axis where 0≤zr <1,and using Re−eiθ≥

−|eiθ_|₋_{1 , the inequality}_2.10_{reduces to}

Re

⎧ ⎨ ⎩

1keiθpzp−_n∞_p_tn|anAn|zn−∞npt−1n|bnBn|zn

−A

pzp₋∞

npt|anAn|zn∞npt−1|bnBn|zn

⎫ ⎬ ⎭

≥ ⎧ ⎨ ⎩

1kprp−_n∞_p_tn|anAn|rn−∞npt−1n|bnBn|rn

−B

prp₋∞

npt|anAn|rn∞npt−1|bnBn|rn

⎫ ⎬ ⎭

⎧ ⎨ ⎩

p1−αrp−∞_n_p_tn1k−pkα|anAn|rn−C

prp₋∞

npt|anAn|rn∞npt−1|bnBn|rn

⎫ ⎬ ⎭,

(6)

whereAdenoteskeiθαpzp−_n∞_p_t|anAn|zn∞npt−1|bnBn|zn,Bdenoteskαprp−

_∞

npt|anAn|rn∞npt−1|bnBn|rn,andCdenotes

_∞

npt−1n1k pkα|bnBn|rn.

Lettingr → 1−, we obtain

p1−α−%∞_n_p_tn1k−pkα|anAn|∞npt−1

n1k pkα|bnBn|

&

1−∞_n_p_t|anAn|∞npt−1|bnBn|

≥0.

2.12 If the condition2.10does not hold, then the numerator in2.12is negative forrsuﬃciently close to 1. Hence there existsz0 r0in0,1for which2.12is negative. Therefore, it follows

thatf /∈H_Fp, t, α, kand so the proof is complete.

Theorem 2.5. Iff ∈H_Fp, t;α, k, then for|z|r <1,|Apt| ≤ |An| ≤ |Bn|,andApt/0,

_fz_≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1bpt−1rpt−1

p1−α

pt1k−pkαApt

−

pt−11k pkα

pt1k−pkαApt

bpt−1Bpt−1

rp1_; _p₁₋_α_≤₁

1bpt−1rpt−1

1

pt1k−pkαApt

−

pt−11k pkα

pt1k−pkαApt

bpt−1Bpt−1

rp1_; _p₁₋_α_≥₁_,

fz≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1−bpt−1rpt−1

−

p1−α

pt1k−pkαApt

−

pt−11k pkα

pt1k−pkαApt

bpt−1Bpt−1

rp1_; _p₁₋_α_≤₁

1−bpt−1rpt−1

−

1

pt1k−pkαApt

−

pt−11k pkα

pt1k−pkαApt

bpt−1Bpt−1

rp1_; _p₁₋_α_≥₁_.

2.13

(7)

Proof. Supposep1−α≤1. Letf ∈H_Fp, t, α, kand|Apt| ≤ |Bn|. In view of1.13, we get

_fz

zpbpt−1zpt−1−

∞

npt

|an|zn− |bn|zn

≤1bpt−1rpt−1

∞

npt

|an||bn|rp1

≤1bpt−1rpt−1

p1−α

pt1k−pkαApt

× ∞

npt

pt1k−pkαApt

p1−α |an||bn|r

p1

≤1bpt−1rpt−1 p

1−α

pt1k−pkαApt

× ⎛ ⎝∞

npt

n1k−pkαApt

p1−α |an|

n1k pkαApt

p1−α |bn|

⎞ ⎠rp1

≤1bpt−1rpt−1

p1−α

pt1k−pkαApt

× ⎛ ⎝∞

npt

n1k−pkα

p1−α |Anan|

n1k pkα

p1−α |Bnbn|

⎞ ⎠_rp1_.

2.14

UsingTheorem 2.4i, we obtain

_fz_≤

1bpt−1rpt−1 p

1−α

pt1k−pkαApt

×

1−

pt−11k pkα

p1−α bpt−1Bpt−1

rp1

1bpt−1rpt−1

p1−α

pt1k−pkαApt

−

pt−11k pkα

pt1k−pkαApt

bpt−1Bpt−1

rp1_.

2.15

(8)

Corollary 2.6. Iff∈H_Fp, t, α, k, then

w:|wz|<

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1− p1−α

pt1k−pkαApt

−

pt1k−pkα−pt−11k pkαBpt−1bpt−1

pt1k−pkαApt ;

p1−α≤1

1− 1

pt1k−pkαApt

−

pt1k−pkα−pt−11k pkαBpt−1bpt−1

pt1k−pkαApt ;

p1−α≥1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⊂fU. 2.16

Theorem 2.7. SupposeAn/0 npt, npt1, . . .andBn/0 npt−1, npt, . . .. Thenf ∈clcoH_Fp, t, α, kif and only if

fz ∞

npt−1

Xnhnz Yngnz, z∈U, 2.17

where

hpt−1z zp

hnz

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

zp₋ p1−α

n1k−pkα|An|z

n_; _n_p_{t, p}_t₁_{, . . .}_{, p}₁₋_α_≤₁_,

zp₋ 1

n1k−pkα|An|z

n_; _n_p_{t, p}_t₁_{, . . .}_{, p}₁₋_α_≥₁_,

gnz

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

zp p1−α

n1k pkα|Bn|z

n _; _n_p_t₋₁_{, p}_{t, . . .}_{, p}₁₋_α_≤₁

zp 1

n1k pkα|Bn|z

n _; _n_p_t₋₁_{, p}_{t, . . .}_{, p}₁₋_α_≥₁

Xpt−1≡Xp1−

⎛ ⎝∞

npt

Xn

∞

npt−1

Yn

⎞

⎠_{, X}_n_≥₀_{, Y}_n_≥₀_.

2.18

(9)

Proof. Supposep1−α≤1. For functions of the form2.17, we can write

fz zp₋ ∞ npt

p1−α

n1k−pkα|An|Xnz

n ∞

npt−1

p1−α

n1k pkα|Bn|Ynz

n_. _2.19

On the other hand, for 0≤Xp≤1, we obtain

∞

npt

n1k−pkα|An|

p1−α

n1k−pkα|An|Xn

∞

npt−1

n1k pkα|Bn|

p1−α

n1k pkα|Bn|Yn

⎛ ⎝∞

npt

Xn

∞

npt−1

Yn

⎞

⎠₁₋_X_p_≤₁_.

2.20

Thusf∈H_Fp, t, α, k, byTheorem 2.4.

Conversely, suppose thatf∈H_Fp, t, α, k. Then, it followsTheorem 2.4that

|an| ≤ p1−α

n1k−pkα|An|, |bn| ≤

p1−α

n1k pkα|Bn|. 2.21

Setting

Xn

p1−α , Yn

n1k pkα|bnBn|

p1−α , 2.22

and defining

Xp1−

⎛ ⎝∞

npt

Xn

∞

npt−1

Yn

⎞

(10)

whereXp≥0, we obtain

fz zp₋ ∞ npt

|an|zn

∞

npt−1 |bn|zn

zp₋ ∞ npt

p1−αXn

n1k−pkα|An|z

n ∞

npt−1

p1−αYn

n1k pkα|Bn|z n

zp₋ ∞ npt

zp₋_h

nzXn−

∞

npt−1

zp₋_g

nzYn

⎛ ⎝₁₋

⎛ ⎝∞

npt

Xn

∞

npt−1

Yn

⎞ ⎠

⎞

⎠zp ∞ npt

hnzXn

∞

npt−1

gnzYn

Xpzp

∞

npt

Xnhnz

∞

npt−1

Yngnz.

2.24

Thusf can be expressed as2.17. The proof for the casep1−α≥1 is similar and hence is omitted.

Theorem 2.8. The classH_Fp, t, α, kis closed under convex combinations.

Proof. Forj1,2, . . . ,let the functionsfjgiven by

fjz zp−

∞

npt

ajnzn

∞

npt−1

bjnz

n _2.25

are inH_Fp, t, α, k. Also suppose the given fixed harmonic functions are given by

Fjz zp

∞

npt

Ajnzn

∞

npt−1

Bjnz

n_. _2.26

For∞_j₁μj 1, 0≤μj ≤1 the convex combinations offjcan be expressed as

∞

j1

μjfjz zp−

∞

npt

⎛ ⎝∞

j1

μjajn ⎞

⎠zn ∞ npt−1

⎛ ⎝∞

j1

μjbjn ⎞

⎠_zn_.

(11)

Since

∞

npt

n1k−pkαajnAjn

∞

npt−1

n1k pkαbjnBjn

≤ ⎧ ⎨ ⎩

p1−α if p1−α≥1

1 if p1−α≤1,

2.28

2.27yields

∞

npt

n1k−pkα

∞

j1

μjajnAjn

∞

npt−1

n1k pkα

∞

j1

μjbjnBjn

∞

j1

μj

⎧ ⎨ ⎩

∞

npt

n1k−pkαajnAjn

∞

npt−1

n1k pkαbjnBjn ⎫ ⎬ ⎭

≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

p1−α

∞

j1

μj p1−α if p1−α≤1

∞

j1

μj1

if p1−α≥1.

2.29

Thus the coeﬃcient estimate given byTheorem 2.4holds. Therefore, we obtain

∞

j1

μjfjz∈H_Fp, t, α, k. 2.30

Acknowledgment

This present investigation is supported with the Project no. D ¨UBAP-07-02-21 by Dicle University, The committee of the Scientific Research Projects.

References

1 O. P. Ahuja and J. M. Jahangiri, “Multivalent harmonic starlike functions with missing coeﬃcients,”

Mathematical Sciences Research Journal, vol. 7, no. 9, pp. 347–352, 2003.

2 H. ¨O. G ¨uney and O. P. Ahuja, “Inequalities involving multipliers for multivalent harmonic functions,”

Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 5, article 190, pp. 1–9, 2006.

3 O. P. Ahuja and J. M. Jahangiri, “Multivalent harmonic starlike functions,”Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 55, no. 1, pp. 1–13, 2001.

(12)

5 H. Silverman, “Harmonic univalent functions with negative coeﬃcients,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 283–289, 1998.

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

7 O. P. Ahuja, R. Aghalary, and S. B. Joshi, “Harmonic univalent functions associated with k-uniformly starlike functions,”Mathematical Sciences Research Journal, vol. 9, no. 1, pp. 9–17, 2005.