A New Subclass of Harmonic Univalent Functions Associated with Dziok Srivastava Operator

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Volume 2012, Article ID 416875,10pages doi:10.1155/2012/416875

Research Article

A New Subclass of Harmonic Univalent Functions

Associated with Dziok-Srivastava Operator

A. L. Pathak,

1

_{K. K. Dixit,}

2

_{and R. Agarwal}

1

1_{Department of Mathematics, Brahmanand College, The Mall, Kanpur 208004, India} 2_{Department of Mathematics, GIIT, Gwalior 474015, India}

Correspondence should be addressed to A. L. Pathak,alpathak.bnd@gmail.com

Received 30 March 2012; Accepted 29 October 2012

Academic Editor: Attila Gil´anyi

Copyrightq2012 A. L. Pathak 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.

The purpose of the present paper is to study a certain subclass of harmonic univalent functions

associated with Dziok-Srivastava operator. We obtain coeﬃcient conditions, distortion bounds,

and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.

1. Introduction

A continuous complex-valued functionfuivdefined in a simply connected domainD, is said to be harmonic inDif bothuandvare harmonic inD. In any simply connected domain

Dwe can writefhg, wherehandgare analytic inD. A necessary and suﬃcient condition forf to be locally univalent and sense preserving in Dis that|hz| > |gz|, z ∈ D. See Clunie and Sheil-Small1.

Denote bySHthe class of functionsf hgthat are harmonic univalent and sense

preserving in the unit diskU{z:|z|<1}for whichf0 fz0−10. Then forfhg∈

SH, we may express the analytic functionhandgas

hz z

∞

k2

akzk, gz

∞

k1

bkzk, |b1|<1. 1.1

Note thatSH, is reduced toS the class of normalized analytic univalent functions if

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For more basic results on harmonic univalent functions, one may refer to the following introductory text book by Duren2 see also3–5.

Forαj ∈ C, j 1,2,3, . . . , qandβj ∈ C− {0,−1,−2,−3, . . .},j 1,2,3, . . . , s, the

generalized hypergeometric functions are defined by

qFs

α1, α2, . . . , αq;β1, β2, . . . , βs;z

∞

k0

α1kα2k· · ·

αq

kzk

β1

k

β2

k· · ·

βs

kk!

,

q≤s1; q, s∈N0{0,1,2, . . .}

,

1.2

whereα_kis the Pochhammer symbol defined by

α_k

⎧ ⎨ ⎩

Γαk

Γα αα1· · ·αk−1, if k1,2,3, . . . ,

1, if k0.

1.3

Corresponding to the function

hα1, α2, . . . , αq;β1, β2, . . . , βs;z

zqFs

α1, α2, . . . , αq;β1, β2, . . . , βs;z

. 1.4

The Dziok-Srivastava operator6,7Hq,sα1, α2, . . . , αq;β1, β2, . . . , βsis defined forf∈

SbyHq,sα1, α2, . . . , αq;β1, β2, . . . , βsfz hα1, α2, . . . , αq;β1, β2, . . . , βs;z∗fz

z

∞

k2

α1k−1α2k−1. . .

αq

k−1

β1

k−1

β2

k−1. . .

βs

k−1

ak z k

k−1!, 1.5

where∗stands for convolution of two power series. To make the notation simple, we write

Hq,sα1fz Hq,s

α1, α2, . . . , αq;β1, β2, . . . , βs

fz. 1.6

Special cases of the Dziok-Srivastava operator includes the Hohlov operator8, the Carlson-Shaﬀer operatorLa, c 9, the Ruscheweyh derivative operator Dn ₁₀_{, and the}

Srivastava-Owa fractional derivative operators11–13.

We define the Dziok-Srivastava operator of the harmonic functionsf hggiven by

1.1as

Hq,sα1fz Hq,sα1hz Hq,sα1gz. 1.7

Recently, Porwal 14, Chapter 5 defined the subclass MHβ ⊂ SH consisting of

harmonic univalent functionsfzsatisfying the following condition:

MH

β

f∈SH: Re

zhz−zgz hz gz

< β

,

1< β≤ 4

3

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He proved that iffhgis given by1.1and if

∞

k2

k−β

β−1|ak|

∞

k1

kβ

β−1|bk| ≤1,

1< β≤ 4

3

. 1.9

Forg≡0 the class ofMHβis reduced to the classMβstudied by Uralegaddi et al.

15.

Generalizing the classMHβ, we letMHα1, βdenote the family of functionsf hg

of form1.1which satisfy the condition

Re

⎧ ⎨ ⎩

zHq,sα1hz

−zHq,sα1gz

Hq,sα1hz Hq,sα1gz ⎫ ⎬

⎭< β, 1.10

where1< β≤4/3andΓα1, k |α1k−1α2k−1· · ·αq_k₋₁/β1k−1β2k−1· · ·βsk−1|.

Further, letMHα1, βbe the subclass ofMHα1, βconsisting of functions of the form

fz z

∞

k2

|ak|zk−

∞

k1

|bk|zk. 1.11

In this paper, we give a suﬃcient condition for f hg, given by 1.1 to be in

MHα1, β, and it is shown that this condition is also necessary for functions inMHα1, β.

We then obtain distortion theorem, extreme points, convolution conditions, and convex combinations and discuss a class preserving integral operator for functions inMHα1, β.

2. Main Results

First, we give a suﬃcient coeﬃcient bound for the classMHα1, β.

Theorem 2.1. Iff hg∈SHis given by1.1and if

∞

k2

k−β

β−1

Γα1, k

k−1!|ak|

∞

k1

kβ

β−1

Γα1, k

k−1!|bk| ≤1, 2.1

thenf ∈MHα1, β.

Proof. Let

∞

k2

k−β

β−1

Γα1, k

k−1!|ak|

∞

k1

kβ

β−1

Γα1, k

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It suﬃces to show that

zHq,sα1hz

−zHq,sα1gz

/Hq,sα1hz Hq,sα1gz

−1

zHq,sα1hz

−zHq,sα1gz

−2β−1

<1,

z∈U.

2.3 We have

zHq,sα1hz

−zHq,sα1gz

−1

zHq,sα1hz

−zHq,sα1gz

−2β−1

≤ _∞

k2k−1A|ak||z|k−1

_∞

k1k1A|bk||z|k−1

2β−1−∞_k2k−2β1A|ak||z|k−1−∞_k1

k2β−1A|bk||z|k−1

≤

_∞

k2k−1A|ak|

_∞

k1k1A|bk|

2β−1−∞_k2k−2β1A|ak| −∞_k1

k2β−1A|bk|

,

2.4

whereAdenotesΓα1, k/k−1!.

The last expression is bounded above by 1, if

∞

k2

k−1Γα1, k

k−1!|ak|

∞

k1

k1Γα1, k

k−1!|bk| ≤2

β−1−

∞

k2

k−2β1Γα1, k

k−1!|ak|

−∞

k1

k2β−1Γα1, k

k−1!|bk|,

2.5

which is equivalent to

∞

k2

k−β

β−1

Γα1, k

k−1!|ak|

∞

k1

kβ

β−1

Γα1, k

k−1!|bk| ≤1. 2.6

But2.6is true by hypothesis. Hence

zHq,sα1hz

₋_z_H

q,sα1gz

_/_H

q,sα1hz Hq,sα1gz

−1

zHq,sα1hz

₋_z_H

q,sα1gz

_/_H

q,sα1hz Hq,sα1gz

−2β−1

<1,

z∈U,

2.7

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In the following theorem, it is proved that the condition2.1is also necessary for functionsf hg ∈MHα1, βthat are given by1.11.

Theorem 2.2. A functionfz z_k2∞ |ak|zk−∞_k1|bk|zk is inMHα1, β, if and only if

∞

k2

k−βΓα1, k

k−1!|ak|

∞

k1

kβΓα1, k

k−1!|bk| ≤

β−1. 2.8

Proof. SinceMHα1, β⊂MHα1, β, we only need to prove the “only if” part of the theorem.

For this we show thatf /∈MHα1, βif the condition2.8does not hold.

Note that a necessary and suﬃcient condition for f hg given by 1.9 is in

MHα1, βif

Re

⎧ ⎨ ⎩

zHq,sα1hz

₋

zHq,sα1gz

Hq,sα1hz Hq,sα1gz ⎫ ⎬

⎭< β 2.9

is equivalent to

Re

β−1z−∞_k2k−βΓα1, k/k−1!|ak|zk−∞_k1

kβΓα1, k/k−1!|bk|zk

z∞_k2Γα1, k/k−1!|ak|zk−

_∞

k1Γα1, k/k−1!|bk|zk

≥0.

2.10

The above condition must hold for all values of z,|z| r < 1. Upon choosing the values ofzon the positive real axis where 0≤zr <1, we must have

β−1−∞_k2k−βΓα1, k/k−1!|ak|rk−1−∞_k1

kβΓα1, k/k−1!|bk|rk−1

1∞_k2Γα1, k/k−1!|ak|rk−1−

_∞

k1Γα1, k/k−1!|bk|rk−1

≥0.

2.11

If the condition2.8does not hold then the numerator of2.11is negative forrand suﬃciently close to 1. Thus there exists a z0 r0 in0,1for which the quotient in 2.11

is negative. This contradicts the required condition for f ∈ MHα1, βand so the proof is

complete.

Next, we determine the extreme points of the closed convex hulls of MHα1, β,

denoted by clcoMHα1, β.

Theorem 2.3. f ∈clcoMHα1, β, if and only if

fz

∞

k1

xkhkz ykgkz

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where

h1z z,

hkz z

β−1

k−β

k−1!

Γα1, k

zk, k 2,3, . . .,

gkz z−

β−1

kβ

k−1!

Γα1, k

zk, k 1,2,3, . . .,

∞

k1

xkyk

1,

2.13

xk≥0, andyk≥0. In particular the extreme points ofMHα1, βare{hk}and{gk}.

Proof. For functionsfof the form2.12, we have

fz

∞

k1

xkhkz ykgkz

z

∞

k2

β−1

k−β

k−1!

Γα1, k

xkzk−

∞

k1

β−1

kβ

k−1!

Γα1, k

ykzk.

2.14

Then

∞

k2

k−β

β−1

Γα1, k

k−1!

β−1

k−β

k−1!

Γα1, k

xk

∞

k1

kβ

β−1

Γα1, k

k−1!

β−1

kβ

k−1!

Γα1, k

yk

∞

k2

xk

∞

k1

yk

1−x1 ≤1,

2.15

and sof ∈clcoMHα1, β.

Conversely, suppose that f ∈ clcoMHα1, β. Set xk k − β/β − 1k −

1!/Γα1, k|ak|,k 2,3,4, . . . and yk k β/β − 1k − 1!/Γα1, k|bk|,k

1,2,3, . . ..

Then note that by Theorem 2.2, 0 ≤ xk ≤ 1, k 2,3,4, . . . and 0 ≤ yk ≤ 1, k

1,2,3, . . .. We definex11−

_∞

k2xk−

_∞

k1yk, and byTheorem 2.2,x1≥0.

Consequently, we obtainfz ∞_k1xkhkz ykgkzas required.

Theorem 2.4. Iff ∈MHα1, β, then

_f_z_≤₁_|_b₁_|_r 1

Γα1,2

β−1

2−β−

β1

2−β|b1|

r2, |z|r <1,

fz≥1− |b1|r− 1

Γα1,2

β−1

2−β−

β1

2−β|b1|

r2, |z|r <1.

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

and will be omitted. Letf∈MHα1, β. Taking the absolute value off, we have

fz≤1|b1|r

∞

k2

|ak||bk|rk

≤1|b1|r

∞

k2

|ak||bk|r2

1|b1|r

β−1

2−β

1

Γα1,2

∞

k2

2−βΓα1,2

β−1 |ak|

2−βΓα1,2

β−1 |bk|

r2

≤1|b1|r

β−1

2−β

1

Γα1,2

∞

k2

k−β

β−1

Γα1, k

k−1!|ak|

kβ

β−1

Γα1, k

k−1!|bk|

r2

≤1|b1|r

β−1

2−β

1

Γα1,2

1−

β1

β−1|b1|

r2

1|b1|r 1

Γα1,2

β−1

2−β−

β1

2−β|b1|

r2.

2.17

For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form

fz z

∞

k2

|ak|zk−

∞

k1

|bk|zk,

Fz z

∞

k2

|Ak|zk−

∞

k1

|Bk|zk.

2.18

We define the convolution of two harmonic functionsfandFas

f∗Fz fz∗Fz z

∞

k2

|akAk|zk−

∞

k1

|bkBk|zk. 2.19

Using this definition, we show that the classMHα1, βis closed under convolution.

Theorem 2.5. For 1< γ ≤β≤4/3, letf∈MHα1, γandF∈MHα1, β. Then

f∗F∈MH

α1, γ

⊆MH

α1, β

. 2.20

Proof. Letfz z∞_k2|ak|zk−∞_k1|bk|zkbe inMHα1, γ, andFz z∞_k2|Ak|zk−

_∞

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Then the convolutionf∗Fis given by2.19. We wish to show that the coeﬃcients of

f∗Fsatisfy the required condition given inTheorem 2.2. ForFz∈MHα1, βwe note that

|Ak| ≤1 and|Bk| ≤1. Now, for the convolution functionf∗F, we obtain

∞

k2

k−α

α−1

Γα1, k

k−1!|akAk|

∞

k1

kα

α−1

Γα1, k

k−1!|bkBk|

≤∞

k2

k−α

α−1

Γα1, k

k−1!|ak|

∞

k1

kα

α−1

Γα1, k

k−1!|bk|

≤1, sincef ∈MH

α1, γ

.

2.21

Thereforef∗F ∈MHα1, γ⊆MHα1, β.

Theorem 2.6. The classMHα1, βis closed under convex combination.

Proof. Fori1,2,3, . . ., letfiz∈MHα1, β, wherefizis given by

fiz z

∞

k2

|aki|zk−

∞

k1

|bki|zk. 2.22

Then byTheorem 2.2,

∞

k2

k−β

β−1

Γα1, k

k−1!|aki|

∞

k1

kβ

β−1

Γα1, k

k−1!|bki| ≤1. 2.23

For∞_i₋₁ti1, 0≤ti≤1, the convex combination offimay be written as

∞

i−1

tifiz z

∞

k2

∞

i−1

ti|aki|

zk₋∞ k1

∞

i−1

ti|bki|

zk. 2.24

Then by2.23,

∞

k2

k−β

β−1

Γα1, k

k−1!

∞

i−1

ti|aki|

∞

k1

kβ

β−1

Γα1, k

k−1!

∞

i−1

ti|bki|

∞

i−1

ti

∞

k2

k−β

β−1

Γα1, k

k−1!|aki|

∞

k1

kβ

β−1

Γα1, k

k−1!|bki|

≤∞

i−1

ti 1.

2.25

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3. A Family of Class Preserving Integral Operator

Letfz hz gzbe defined by1.1; thenFzdefined by the relation

Fz c1 zc

z

0

tc−1htdtc1 zc

z

0

tc−1_g_t_dt, _{c >}−₁_. 3.1

Theorem 3.1. Letfz hz gz∈SHbe given by1.11andf ∈MHα1, β, thenFzis

defined by3.1also belong toMHα1, β.

Proof. Letfz z∞_k2|ak|zk−

_∞

k1|bk|zkbe inMHα1, β, then byTheorem 2.2, we have

∞

k2

k−β

β−1

Γα1, k

k−1!|ak|

∞

k1

kβ

β−1

Γα1, k

k−1!|bk| ≤1. 3.2

By definition ofFz, we have

Fz z

∞

k2

c1

ck|ak|z

k₋∞

k1

c1

ck|bk|z

k_.

3.3

Now we have

∞

k2

k−β

β−1

Γα1, k

k−1!

c1

ck|ak|

∞

k1

kβ

β−1

Γα1, k

k−1!

c1

ck|bk|

≤∞

k2

k−β

β−1

Γα1, k

k−1!|ak|

∞

k1

kβ

β−1

Γα1, k

k−1!|bk|

≤1.

3.4

ThusFz∈MHα1, β.

Acknowledgments

The first author is thankful to University Grants Commission, New Delhi Government of India for supporting this paper financially under research project no. 8-2223/2011

MRP/Sc/NRCB. The authors are also thankful to the reviewers for giving fruitful suggestions and comments.

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A New Subclass of Harmonic Univalent Functions Associated with Dziok Srivastava Operator

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