On Harmonic Functions Defined by Derivative Operator

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Journal of Inequalities and Applications Volume 2008, Article ID 263413,10pages 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,[email protected]

Received 16 September 2007; Revised 20 November 2007; Accepted 26 November 2007

Recommended by Vijay Gupta

Let SH 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∞_k₂akzk, gz k∞1bkzk|b1| <

1. In this paper, we introduce the classMHn, λ, αof functionsfh

––

gwhich are harmonic inU.

A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also nec-essary for the classM––

Hn, λ, αiffnz h

––

gn∈MHn, λ, α, wherehz z−∞k2|ak|zk, gnz

−1n∞

k1|bk|zkandn ∈ N0. Coeﬃcient conditions, such as distortion bounds, convolution

con-ditions, convex combination, extreme points, and neighborhood for the classM––

Hn, λ, α, are

ob-tained.

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 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 andgthe coanalytic part off. A necessary and suﬃcient condition forf to be locally univalent and sense-preserving in Dis that|hz|>|gz|inD; see2.

Denote bySH the class of functionsf hg that are harmonic, univalent, and

sense-preserving in the unit diskU {z : |z| < 1}for whichf0 h0 fz0−1 0. Then for

f hg ∈ SH, we may express the analytic functionshandgas

hz z∞

k2

akzk, gz

∞

k1

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Observe thatSHreduces toS, the class of normalized univalent analytic functions, if the

coan-alytic part offis zero. Also, denote byS∗_H the subclasses ofSH consisting of functionsf that

mapUonto starlike domain.

Forf hg given by 1.1, we define the derivative operator introduced by authors

see1offas

Dn

λfz Dnλhz −1nDnλgz, n, λ∈N0 N∪ {0}, z∈U, 1.2

whereDn_λhz z∞_k₂knCλ, kakzk, Dn_λgz ∞k1knCλ, kbkzk, andCλ, k kλλ−1.

We letMHn, λ, αdenote the family of harmonic functionsfof the form1.1such that

Re

_Dn1

λ fz

Dn

λfz

> α, 0≤α <1, 1.3

whereDn_λfis defined by1.2.

If the coanalytic part off hgis identically zero, then the classMHn, λ, αturns out

to be the classRn_λαintroduced by Al-Shaqsi and Darus1for the analytic case.

LetM_Hn, λ, αdenote that the subclass ofMHn, λ, αconsists of harmonic functions

fnhgnsuch thathandgnare of the form

hz z−∞

k2

_a_k_zk_, _g

nz −1n

∞

k1

_b_k_zk_. _1.4

It is clear that the class MHn, λ, αincludes a variety of well-known subclasses of SH. For

example,MH0,0, α≡ S∗_Hαis 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 ≡

M_H0,1, α≡ HKαis the class of sense-preserving, harmonic, univalent functionsf which are convex of order α in U, that is, ∂/∂θarg∂/∂θfreiθ > α. Note that the classes

S∗

H and HKαwere introduced and studied by Jahangiri3. Also we notice that the class

M_Hn,0, αis the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al.4; andM_H0, λ, αis the class of Ruscheweyh-type harmonic univalent functions stud-ied by Murugusundaramoorthy and Vijaya5.

In 1984, Clunie and Sheil-Small2investigated the classSHas 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 Silvia7, and Jahangiri3,8

studied the harmonic univalent functions. Jahangiri and Silverman 9 prove the following theorem.

Theorem 1.1. Letf hg given by1.1. If

∞

k2

k akbk≤1−b1, 1.5

thenfis sense-preserving, harmonic, and univalent inUandf ∈S∗_Hconsists of functions inSHwhich

are starlike inU.

The condition1.5is also necessary iff∈ TH ≡M_H0,0,0.

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also necessary for functions in the classM_Hn, λ, α. Also, we obtain distortion theorems and characterize the extreme points for functions inM_Hn, λ, α. Closure theorems and application of neighborhood are also obtained.

2. Coefficient bounds

We begin with a suﬃcient coeﬃcient condition for functions inMHn, λ, α. Theorem 2.1. Letf hg be given by1.1. If

∞

k1

k−αak kαbkknCλ, k≤21−α, 2.1

wherea1 1, n, λ ∈ N0, Cλ, k k_λλ−1, and0 ≤ α < 1, thenf is sense-preserving, harmonic,

univalent inU, andf ∈MHn, λ, α.

Proof. Ifz1/z2, then

f z1

−f z2

h z1

−h z2

≥1−g z1

−g z2

h z1

−h z2

1−

_∞

k1bk zk1−z k 2

z1−z2

∞k2ak zk1−zk2

>1−

_∞

k1kbk 1−∞_k₂kak

≥1−

_∞

k1 kαknCλ, k/1−αbk

1−∞k2 k−αknCλ, k/1−αak ≥0,

2.2

which proves univalence. Note thatf is sense-preserving inU. This is because

h_z_≥₁₋∞

k2

kak|z|k−1

>1−

∞

k2

k−αkn_C_{λ, k}

1−α ak

≥∞

k1

kαkn_C_{λ, k}

1−α bk

>∞

k1

kαkn_C_{λ, k}

1−α bk|z|

k−1_≥∞

k1

kbk|z|k−1≥gz.

2.3

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

₁₋αDn

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SubstitutingDn_λfzin2.4yields, by2.1, we obtain

₁₋αDn

λfz Dnλ1fz−1αDλnfz− Dλn1fz

2−αz

∞

k2

k1−αknCλ, kakzk−−1n

∞

k1

k−1αknCλ, kbkzk

−−αz∞

k2

k−1−αknCλ, kakzk −−1n

∞

k1

k1αknCλ, kbkzk

≥21−α|z|

1−∞

k2

1−α ak|z|

k−1∞

k1

kαkn_C_{λ, k}

1−α bk|z|

k−1

≥21−α

1−

∞

k2

1−α ak−

∞

k1

kαkn_C_{λ, k}

1−α bk

.

2.5

This last expression is nonnegative by2.1, and so the proof is complete.

The harmonic function

fz z∞

k2

1−α

k−αkn_C_{λ, k}xkzk

∞

k1

1−α

kαkn_C_{λ, k}ykzk, 2.6

wheren, λ∈ N0and∞k2|xk|

_∞

k1|yk| 1 show that the coeﬃcient bound given by2.1is sharp. The functions of the form2.6are inMHn, λ, αbecause

∞

k1

_k₋_α

1−αak

kα

1−αbk

knCλ, k 1

∞

k2

_x_k∞

k1

_y_k₂_. _2.7

In the following theorem, it is shown that the condition 2.1is also necessary for functions

fnhgn, wherehandgnare of the form1.4.

Theorem 2.2. Letfnhgnbe given by1.4. Thenfn∈M_Hn, λ, αif and only if

∞

k1

k−αak kαbkknCλ, k≤21−α, 2.8

wherea11, n, λ∈N0, Cλ, k k_λλ−1, and0≤α <1.

Proof. SinceM_Hn, λ, α⊂ MHn, λ, α, we only need to prove the “if and only if” part of the

theorem. To this end, for functions fn of the form1.4, we notice that the condition 1.3is

equivalent to

Re

1−αz−∞_k₂k−αknCλ, kakzk−−12n∞k1kαknCλ, kbkzk

z−∞k2knCλ, kakzk −12n

_∞

k1knCλ, kbkzk

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The above required condition2.9must hold for all values ofzinU. Upon choosing the values ofzon the positive real axis, where 0≤zr <1, we must have

1−α−∞k2k−αknCλ, kakrk−1−

_∞

k1kαknCλ, kbkrk−1 1−∞k2knCλ, kakrk−1

_∞

k1knCλ, kbkrk−1

≥0. 2.10

If the condition2.8does not hold, then the numerator in2.10is negative forr suﬃciently close to 1. Hence there exist z0 r0 in0,1for which the quotient in2.8is negative. This contradicts the required condition forfn∈M_Hn, λ, αand so the proof is complete.

3. Distortion bounds

In this section, we will obtain distortion bounds for functions inM_Hn, λ, α.

Theorem 3.1. Letfn∈M_Hn, λ, α. Then for|z|r <1, one has

fnz≤ 1b1r 1 2nλ1

1−α 2−α−

1α 2−αb1

r2_,

_f_n_z_≥ ₁₋_b₁_r₋ 1 2nλ1

1−α 2−α−

1α 2−αb1

r2_.

3.1

Proof. We only prove the left-hand inequality. The proof for the right-hand inequality is similar

and will be omitted. Letfn∈M_Hn, λ, α. Taking the absolute value offn, we obtain

fnz

z−

∞

k2

akzk −1n

∞

k1

bkzk

≥ 1−b1r−

∞

k2

akbkrk

≥ 1−b1r−r2

∞

k2

akbk

≥ 1−b1r− 1−

α

2−α2nλ1

_∞

k2

2−α2nλ1

1−α ak

2−α2nλ1

1−α bk

r2

≥ 1−b1r− 1−

α

2−α2nλ1

_∞

k2

1−α ak

kαkn_C_{λ, k}

1−α bk

r2

≥ 1−b1r−

1−α

2−α2nλ1

1−1α 1−αb1

r2_.

3.2

The functions

fz zb1z 1 2nλ1

1−α 2−α−

1α 2−αb1

z2_,

fz 1−b1z− 1 2nλ1

1−α 2−α−

1α 2−αb1

z2

3.3

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The following covering result follows from the left-hand inequality inTheorem 3.1.

Corollary 3.2. If the functionfnhgn, wherehandggiven by1.4are inM_Hn, λ, α, then

w:|w|< 2

n1_λ₁₋₁₋ ₂n_λ₁₋₁_α

2nλ12−α −

2n1_λ₁₋₁₋ ₂n_λ₁ ₁_α 2nλ12−α b1

⊂fnU.

3.4

4. Convolution, convex combination, and extreme points

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

For harmonic functions fnz z − k∞2akzk −1

n∞

k1bkz

k _and _F

nz z −

_∞

k2Akzk −1

n∞

k1Bkz k

, the convolution offnandFnis given by

fn∗Fnz fnz∗Fnz z−

∞

k2

akAkzk −1n

∞

k1

bkBkzk. 4.1

Theorem 4.1. For 0 ≤ β ≤ α < 1, let fn ∈ M_Hn, λ, α and Fn ∈ M_Hn, λ, β. Then fn∗Fn

∈M_Hn, λ, α⊂M_Hn, λ, β.

Proof. We wish to show that the coeﬃcients of fn∗Fn satisfy the required condition given in

Theorem 2.2. ForFn∈M_Hn, λ, β,we note that|Ak| ≤1 and|Bk| ≤1. Now, for the convolution

functionfn∗Fn, we obtain

∞

k2

k−βkn_C_{λ, k}

1−β akAk

∞

k1

kβkn_C_{λ, k}

1−β bkBk

≤∞

k2

k−βkn_C_{λ, k}

1−β ak

∞

k1

kβkn_C_{λ, k}

1−β bk

≤∞

k2

1−α ak

∞

k1

kαkn_C_{λ, k}

1−α bk≤1,

4.2

since 0≤β≤α <1 andfn∈M_Hn, λ, α. Thereforefn∗Fn∈M_Hn, λ, α⊂M_Hn, λ, β.

We now examine the convex combination ofM_Hn, λ, α. Let the functionsfnjzbe defined, forj 1,2, . . . ,by

fnjz z−

∞

k2

ak,jzk −1n

∞

k1

bk,jzk. 4.3

Theorem 4.2. Let the functions fnjz defined by 4.3 be in the class M_Hn, λ, α for every j 1,2, . . . , m. Then the functionstjzdefined by

tjz m

j1

cjfnjz, 0≤cj ≤1 4.4

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Proof. According to the definition oftj, we can write

tjz z−

∞

k2

_m

j1

cjak,j

zk ₋₁n∞ k1

_m

j1

cjbn,j

zk. 4.5

Further, sincefnjzare inM_Hn, λ, αfor everyj1,2, . . ., then by2.8, we have

∞

k1

k−α

_m

j1

cjak,j

kα _m

j1

cjbk,j

kn_C_{λ, k}

m

j1

cj

_∞

k1

k−αan,j kαbn,jknCλ, k

≤m

j1

cj21−α≤21−α.

4.6

Hence the theorem follows.

Corollary 4.3. The classM_Hn, λ, αis closed under convex linear combination.

Proof. Let the functionsfnjz j 1,2defined by4.1be in the classM_Hn, λ, α. Then the

functionΨzdefined by

Ψz μfn1z 1−μfn2z, 0≤μ≤1 4.7

is in the classM_Hn, λ, α. Also, by takingm 2,t1 μ, andt2 1−μinTheorem 4.1, we have the corollary.

Next we determine the extreme points of closed convex hulls ofM_Hn, λ, αdenoted by clcoM_Hn, λ, α.

Theorem 4.4. Letfnbe given by1.4. Thenfn∈M_Hn, λ, αif and only if

fnz

∞

k1

Xkhkz Ykgnkz

, 4.8

where h1z z, hkz z − 1 − α/k − αknCλ, kzk, k 2,3, . . . , gnkz z

−1n1−α/kαknCλ, kzk, k 1,2,3, . . . ,and ∞k1 Xk Yk

1, Xk ≥ 0, Yk ≥ 0. In

particular, the extreme points ofM_Hn, λ, αarehkandgnk

.

Proof. For the functionsfnof the form4.8, we have

fnz

∞

k1

Xkhkz Ykgnkz

∞

k1

XkYkz−

∞

k2

1−α

k−αkn_C_{λ, k}Xkzk −1n

∞

k1

1−α

kαkn_C_{λ, k}Ykzk.

4.9

Then

∞

k2

1−α ak

∞

k1

kαkn_C_{λ, k}

1−α bk

∞

k2

Xk

∞

k1

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and sofn∈clcoM_Hn, λ, α.

Conversely, suppose thatfn∈clcoM_Hn, λ, α. Setting

Xk k−αk

n_C_{λ, k}

1−α ak, 0≤Xk ≤1, k 2,3, . . . ,

Yk kαk

n_C_{λ, k}

1−α bk, 0≤Yk ≤1, k1,2,3, . . . ,

4.11

andX11−∞k2Xk−

_∞

k1Yk. Therefore,fncan be written as

fnz z−

∞

k2

akzk −1n

∞

k1

bkzk

z−∞

k2

1−αXk

k−αkn_C_{λ, k}zk −1n

∞

k1

1−αYk

kαkn_C_{λ, k}zk

z∞

k2

hkz−zXk

∞

k1

gnkz−z

Yk

∞

k2

hkzXk

∞

k1

gnkzYkz

1−

∞

k2

Xk−

∞

k1

Yk

∞

k1

hkzXkgnkzYk

, as required.

4.12

Using Corollary 4.3 we have clcoM_Hn, λ, α M_Hn, λ, α. Then the statement of Theorem 4.4is really forf ∈M_Hn, λ, α.

5. An application of neighborhood

In this section, we will prove that the functions in a neighborhood ofM_Hn, λ, αare starlike harmonic functions.

Following10, we defined theδ-neighborhood of a functionf∈ THby

Nδf

Fz z−∞

k2

Akzk−

∞

k1

Bkzk,

∞

k2

kak−Akbk−Bkb1−B1≤δ

,

5.1

whereδ >0.

Theorem 5.1. Let

δ 2−α2

n_λ

1−1α− 2−α2nλ1−1−αb1

2−α2nλ1 . 5.2

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Proof. Supposefn ∈ M_Hn, λ, α. LetFn HGn ∈ Nδ fn,where H z−∞k2Akzk and

Gn −1n∞k1Bkzk. We need to show thatFn ∈ TH. In other words, it suﬃces to show that

Fnsatisfies the conditionTF ∞k2k|Ak||Bk| |B1| ≤1. We observe that

TF ∞

k2

kAkBkB1

∞

k2

kAk−akakBk−bkbkB1−b1b1

∞

k2

kAk−akBk−bk

∞

k2

kakbkB1−b1b1

_∞

k2

kAk−akBk−bkB1−b1

∞

k2

kakbkb1

δb1

∞

k2

kakbk

δb1 1−

α

2−α2nλ1

∞

k2

2−α 1−αak

2α 1−αbk

2nλ1

≤δb1

1−α

2−α2nλ1

∞

k2

_k₋_α

1−αak

kα

1−αbk

kn_C_{λ, k}

≤δb1 1−α

2−α2nλ1

1−1α 1−αb1

.

5.3

Now this last expression is never greater than one if

δ≤1−b1− 1−

α

2−α2nλ1

1−1α 1−αb1

2−α2

n_λ₁₋₁_α₋ ₂₋_α₂n_λ₁₋₁₋_α_b

1

2−α2nλ1 .

5.4

Acknowledgment

The work presented here was supported by Fundamental Research Grant Scheme UKM-ST-01-FRGS0055-2006.

References

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3J. M. Jahangiri, “Harmonic functions starlike in the unit disk,”Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999.

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5G. 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.

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