Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions

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Volume 2010, Article ID 369078,15pages doi:10.1155/2010/369078

Research Article

Differential Subordination Defined

by New Generalised Derivative Operator

for Analytic Functions

Ma’moun Harayzeh Al-Abbadi and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor (Darul Ehsan), Malaysia

Correspondence should be addressed to Maslina Darus,[email protected]

Received 9 June 2009; Revised 20 December 2009; Accepted 9 March 2010

Academic Editor: Dorothy Wallace

Copyrightq2010 M. Harayzeh Al-Abbadi 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.

A new generalised derivative operator μn,m_λ

1,λ2 is introduced. This operator generalised many well-known operators studied earlier by many authors. Using the technique of diﬀerential subordination, we will study some of the properties of diﬀerential subordination. In addition we investigate several interesting properties of the new generalised derivative operator.

1. Introduction and Preliminaries

LetAdenote the class of functions of the form

fz z

∞

k2

akzk, whereakis complex number, 1.1

which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane

C. Let S, S∗α, Cα 0 ≤ α < 1 denote the subclasses of A consisting of functions that are univalent, starlike of orderα,and convex of orderαinU, respectively. In particular, the classesS∗0 S∗ andC0 Care the familiar classes of starlike and convex functions inU, respectively. A functionf ∈Cαif Re1zf/f> α.Furthermore a functionf analytic in

Uis said to be convex if it is univalent andfUis convex.

LetHUbe the class of holomorphic function in unit discU{z ∈C: |z|<1}. Let

An

f∈ HU: fz zan1zn1· · ·, z∈U

, 1.2

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Fora∈Candn∈N{1,2,3, . . .}we let

Ha, n f ∈ HU: fz zanznan1zn1· · ·, z∈U

. 1.3

Letfz z∞_k₂akzk and gz z

_∞

k2bkzk be analytic in the open unit discU{z∈ C: |z|<1}. Then the Hadamard productor convolutionf∗gof the two functionsf,gis defined by

fz∗gz f∗gz z

∞

k2

akbkzk. 1.4

Next, we state basic ideas on subordination. Iff andgare analytic inU, then the functionf

is said to be subordinate tog, and can be written as

f≺g, fz≺gz, z∈U, 1.5

if and only if there exists the Schwarz functionw, analytic inU, withw0 0 and|wz|<

1 such thatfz gwz,z∈U.

Furthermore ifg is univalent inU, thenf ≺ gif and only iff0 g0andfU ⊂ gU see1, page 36.

Letψ : C3_×_U _→ _C_{and let}_h_{be univalent in}_U_{. If}_p_{is analytic in}_U_{and satisfies the}

second-orderdiﬀerential subordination

ψpz, zpz, z2pz;z ≺hz, z∈U, 1.6

thenpis called a solution of the diﬀerential subordination.

The univalent function q is called a dominant of the solutions of the diﬀerential subordination, or more simply a dominant, if p ≺ qfor allp satisfying1.6. A dominant

qthat satisfiesq≺qfor all dominantsqof1.6is said to be the best dominant of1.6.Note that the best dominant is unique up to a rotation ofU.

Now,x_kdenotes the Pochhammer symbolor the shifted factorialdefined by

x_k

⎧ ⎨ ⎩

1 fork0, x∈C\ {0},

xx1x2· · ·xk−1 fork∈N{1,2,3, . . .}, x∈C. 1.7

To prove our results, we need the following equation throughout the paper:

μn,m_λ1,λ21fz 1−λ1

μn,m_λ1,λ2fz∗φλ2z

λ1z

, z∈U, 1.8

wheren, m∈N0{0,1,2, . . .},λ2 ≥λ1≥0, andφλ2zis analytic function given by

φλ2z z ∞

k2 zk

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Hereμn,m_{λ1, λ2} is the generalized derivative operator which we shall introduce later in the paper. Moreover, we need the following lemmas in proving our main results.

Lemma 1.1see2, page 71. Lethbe analytic, univalent, and convex inU, withh0 a,γ /0

and, Re γ ≥0. Ifp∈ Ha, nand

pz zp

_z

γ ≺hz, z∈U, 1.10

then

pz≺qz≺hz, z∈U, 1.11

whereqz γ/nzγ/nz

0 httγ/n−1dt,z∈U.

The functionqis convex and is the besta, n-dominant.

Lemma 1.2see3. Letgbe a convex function inUand let

hz gz nαzgz, 1.12

whereα >0 andnis a positive integer.

If

pz g0 pnznpn1zn1· · ·, z∈U, 1.13

is holomorphic inUand

pz αzpz≺hz, z∈U, 1.14

then

pz≺gz, 1.15

and this result is sharp.

Lemma 1.3see4. Letf∈ A, if

Re

1zf

_z

fz

>−1

2, 1.16

then

2

z

0

ftdt, z∈U, z /0, 1.17

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2. Main Results

In the present paper, we will use the method of diﬀerential subordination to derive certain properties of generalised derivative operatorμn,m_λ1,λ2fz.Note that diﬀerential subordination has been studied by various authors, and here we follow similar works done by Oros5and G. Oros and G. I. Oros6.

In order to derive our new generalised derivative operator, we define the analytic function

F_λ1,λ2m z z

∞

k2

1λ1k−1m

1λ2k−1m−1

zk, 2.1

where m ∈ N0 {0,1,2, . . .} and λ2 ≥ λ1 ≥ 0. Now, we introduce the new generalised

derivative operatorμn,m_λ1,λ2as follows.

Definition 2.1. Forf∈ Athe operatorμn,m_λ1,λ2is defined byμn,m_λ1,λ2:A → A

μn,m_λ1,λ2fz Fm

λ1,λ2z∗Rnfz, z∈U, 2.2

wheren, m ∈ N0 N∪ {0}, λ2 ≥ λ1 ≥ 0,andRnfzdenotes the Ruscheweyh derivative

operator7, given by

Rnfz z

∞

k2

cn, kakzk, n∈N0, z∈U, 2.3

wherecn, k n1_k₋₁/1_k₋₁.

Iffis given by1.1, then we easily find from equality2.2that

μn,m_λ1,λ2fz z

∞

k2

1λ1k−1m

1λ2k−1m−1

cn, kakzk, z∈U, 2.4

wheren, m∈N0{0,1,2. . .}, λ2 ≥λ1≥0,andcn, k

nk−1

n n1_k₋₁/1_k₋₁.

Remark 2.2. Special cases of this operator include the Ruscheweyh derivative operator in two

cases μn,₀_,λ21 ≡ Rn _and _μn,0

λ1,0 ≡ Rn 7, the Salagean derivative operatorμ 0,m

1,0 ≡ Sn 8, the

generalised Ruscheweyh derivative operator in two cases μn,_λ1,λ21 ≡ Rn λ and μ

n,0

λ1,λ2 ≡ Rnλ 9,

the generalised Salagean derivative operatorμ0_λ1,,m₀ ≡ Sn

β introduced by Al-Oboudi10, and

the generalised Al-Shaqsi and Darus derivative operatorμn,m_λ1,₀ ≡ Dn

λ,β that can be found in 11.

Now, we remind the well-known Carlson-Shaﬀer operatorLa, c 12associated with the incomplete beta functionφa, c;z, defined by

La, c:A → A,

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where

φa, c;z z

∞

k2

a_k₋₁

c_k₋₁z

k_, _2.6

ais any real number, andc /∈z−₀; z−₀ {0,−1,−2, . . .}. It is easily seen that

μ0_λ1,,0₀fz μ₀0,m_,₀fz μ0₀,_,λ21 fz μ1₀,_,2₁fz La, afz fz,

μ1_λ1,,0₀fz μ₀1,m_,₀fz μ1₀,_,λ21 fz μ0_λ1,,0₁fz L2,1fz zfz,

2.7

and also

μ_λ1,a−1₀,0fz μ₀a_,λ2−1,1fz μ₀a−_,₀1,mfz La,1fz, 2.8

wherea1,2,3, . . . .

Next, we give the following.

Definition 2.3. Forn, m ∈ N0, λ2 ≥ λ1 ≥ 0, and 0 ≤ α < 1,let Rn,m_λ1,λ2αdenote the class of

functionsf ∈ Awhich satisfy the condition

Reμn,m_λ1,λ2fz > α, z∈U. 2.9

Also letKn,m_λ1,λ2δdenote the class of functionsf∈ Awhich satisfy the condition

Reμn,m_λ1,λ2fz∗φλ2z

> δ, z∈U. 2.10

Remark 2.4. It is clear thatR0_λ1,,1₀α≡Rλ1, α, and the class of functionsf∈ Asatisfying

Reλ1zfz fz

> α, z∈U 2.11

is studied by Ponnusamy13and others.

Now we begin with the first result as follows.

Theorem 2.5. Let

hz 1 2α−1z

1z , z∈U, 2.12

be convex in U, with h(0)=1 and 0 ≤ α < 1. If n, m ∈ N0, λ2 ≥ λ1 ≥ 0, and the diﬀerential

subordination

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holds, then

≺qz 2α−121−α

λ1z1/λ1 σ

1

λ1

, 2.14

whereσis given by

σx

_z

0 tx−1

1tdt, z∈U. 2.15

The functionqis convex and is the best dominant.

Proof. By diﬀerentiating1.8, with respect toz, we obtain

μn,m_λ1,λ21fz μn,m_λ1,λ2fz∗φλ2z

λ1z

. 2.16

Using2.16in2.13, diﬀerential subordination2.13becomes

λ1z

≺hz 1 2α−1z

1z . 2.17

Let

pz μn,m_λ1,λ2fz∗φλ2z

z ∞ k2

1λ1k−1m

1λ2k−1m

cn, kakzk

1p1zp2z2· · · ,

p∈ H1,1, z∈U.

2.18

Using2.18in2.17, the diﬀerential subordination becomes

pz λ1zpz≺hz

1 2α−1z

1z . 2.19

By usingLemma 1.1, we have

pz≺qz 1 λ1z1/λ1

_z

0

htt1/λ1−1_dt

1

λ1z1/λ1

z

0

1 2α−1t

1t

t1/λ1−1dt

2α−121−α

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whereσis given by2.15, so we get

≺qz 2α−121−α

λ1z1/λ1 σ

1

λ1

. 2.21

The functionqis convex and is the best dominant. The proof is complete.

Theorem 2.6. Ifn, m∈N0, λ2 ≥λ1≥0,and 0≤α <1,then one has

R_λ1,λ2n,m1α⊂K_λ1,λ2n,mδ, 2.22

where

δ2α−121−α

λ1 σ

1

λ1

, 2.23

andσis given by2.15.

Proof. Letf∈ Rn,m_λ1,λ21α, then from2.9we have

Reμn,m_λ1,λ21fz > α, z∈U, 2.24

which is equivalent to

μn,m_λ1,λ21fz ≺hz 1 2α−1z

1z . 2.25

UsingTheorem 2.5, we have

≺ qz 2α−121−α

λ1z1/λ1 σ

1

λ1

. 2.26

Sinceqis convex andqUis symmetric with respect to the real axis, we deduce that

Reμn,m_λ1,λ2fz∗φλ2z

>Re q1 δδα, λ1

2α−121−α

λ1 σ

1

λ1

,

2.27

from which we deduceRn,m_λ1,λ21α⊂Kn,m_λ1,λ2δ.This completes the proof ofTheorem 2.6.

Remark 2.7. Special case ofTheorem 2.6withλ20 was given earlier in11.

Theorem 2.8. Letqbe a convex function inU, withq0 1,and let

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Ifn, m∈N0, λ2≥λ1≥0,andf∈ Aand satisfies the diﬀerential subordination

μn,m_λ1,λ21fz ≺hz, z∈U, 2.29

then

≺qz, z∈U, 2.30

and this result is sharp.

Proof. Using2.18in2.16, diﬀerential subordination2.29becomes

pz λ1zpz≺hz qz λ1zqz, z∈U. 2.31

UsingLemma 1.2, we obtain

pz≺qz, z∈U. 2.32

Hence

≺qz, z∈U. 2.33

And the result is sharp. This completes the proof of the theorem.

We give a simple application forTheorem 2.8.

Example 2.9. Forn 0, m 1, λ2 ≥ λ1 ≥ 0, qz 1z/1−z, f ∈ A,andz ∈ Uand

applyingTheorem 2.8, we have

hz 1z

1−zλ1z

1z

1−z

12λ1z−z2

1−z2 . 2.34

By using1.8we find that

μ0_λ1,λ2,1 fz 1−λ1fz λ1zfz. 2.35

Now,

μ0_λ1,λ2,1 fz∗φλ2z 1−λ1

z

∞

k2

akzk

1λ2k−1

λ1

z

∞

k2

akkzk

1λ2k−1

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A straightforward calculation gives the following:

μ0_λ1,λ2,1 fz∗φλ2z

1

∞

k2

1λ1k−1kak

1λ2k−1 zk−1

z

_∞

k2kak1λ1k−1/1λ2k−1zk z

fz∗φλ2z

∗z∞_k₂k1λ1k−1zk

z .

2.37

Similarly, using1.8, we find that

μ0_λ1,λ2,2 fz 1−λ1

λ1z

, 2.38

then

μ_λ1,λ20,2 fz μ_λ1,λ20,1 fz∗φλ2z

λ1z

. 2.39

By using2.37we obtain

∞ k2

kk−1ak1λ1k−1

1λ2k−1

zk−2_. _2.40

We get

μ0_λ1,λ2,2 fz 1

∞

k2

kak1λ1k−1

1λ2k−1

zk−1_λ 1

∞

k2

kk−1ak1λ1k−1

1λ2k−1

zk−1

1

∞

k2

kak1λ1k−12

1λ2k−1 zk−1

fz∗φλ2z

z .

2.41

FromTheorem 2.8we deduce that

fz∗φλ2z

z ≺

12λ1z−z2

1−z2 ,

2.42

implies that

fz∗φλ2z

z ≺

1z

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Theorem 2.10. Letqbe a convex function inU, withq0 1 and let

hz qz zqz, z∈U. 2.44

Ifn, m∈N0, λ2≥λ1≥0,andf ∈ Aand satisfies the diﬀerential subordination

μn,m_λ1,λ2fz ≺hz, 2.45

then

μn,m_λ1,λ2fz

z ≺qz, z∈U. 2.46

And the result is sharp.

Proof. Let

pz μ

n,m λ1,λ2fz

z

_∞

k2

1λ1k−1m/1λ2k−1m−1 cn, kakzk

z

1p1zp2z2· · ·,

p∈ H1,1, z∈U.

2.47

Diﬀerentiating2.47, with respect toz, we obtain

μn,m_λ1,λ2fz pz zpz, z∈U. 2.48

Using2.48,2.45becomes

pz zpz≺hz qz zqz. 2.49

UsingLemma 1.2, we deduce that

pz≺qz, z∈U, 2.50

and using2.47, we have

μn,m_λ1,λ2fz

z ≺qz, z∈U. 2.51

This provesTheorem 2.10.

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Example 2.11. Forn0, m1, λ2 ≥λ1 ≥0, qz 1/1−z, f ∈ A,andz∈Uand applying

Theorem 2.10, we have

hz 1

1−zz

1 1−z

1

1−z2. 2.52

FromExample 2.9, we have

μ0_λ1,λ2,1 fz 1−λ1fz λ1zfz, 2.53

so

μ0_λ1,λ2,1 fz fz λ1zfz. 2.54

Now, fromTheorem 2.10we deduce that

fz λ1zfz≺

1

1−z2 2.55

implies that

1−λ1fz λ1zfz

z ≺

1

1−z. 2.56

Theorem 2.12. Let

hz 1 2α−1z

1z , z∈U, 2.57

be convex inU, withh0 1 and 0≤α <1. Ifn, m∈N0, λ2≥λ1 ≥0, f ∈ A,and the diﬀerential

subordination holds as

μn,m_λ1,λ2fz ≺hz, 2.58

then

μn,m_λ1,λ2fz

z ≺qz 2α−1

21−αln1z

z . 2.59

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Proof. Let

pz μ

n,m λ1,λ2fz

z

_∞

k2

z

1p1zp2z2· · ·,

p∈ H1,1, z∈U.

2.60

Diﬀerentiating2.60, with respect toz, we obtain

Using2.61, the diﬀerential subordination2.58becomes

pz zpz≺hz 1 2α−1z

1z , z∈U. 2.62

FromLemma 1.1, we deduce that

pz≺qz 1 z

z

0 htdt

1

z

_z

0

1 2α−1t

1t

dt

1

z

0

1

1tdt 2α−1

_z

0 t

1tdt

2α−121−αln1z

z .

2.63

Using2.60, we have

μn,m_λ1,λ2fz

z ≺qz 2α−1

21−αln1z

z . 2.64

The proof is complete.

FromTheorem 2.12, we deduce the following corollary.

Corollary 2.13. Iff∈ Rn,m_λ1,λ2α, then

Re

μn,m_λ1,λ2fz z

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Proof. Sincef∈ Rn,m_λ1,λ2α, fromDefinition 2.3we have

Reμn,m_λ1,λ2fz > α, z∈U, 2.66

which is equivalent to

μn,m_λ1,λ2fz ≺hz 1 2α−1z

1z . 2.67

UsingTheorem 2.12, we have

μn,m_λ1,λ2fz

z ≺qz 2α−1 21−α

ln1z

z . 2.68

Sinceqis convex andqUis symmetric with respect to the real axis, we deduce that

Re

μn,m_λ1,λ2fz z

>Req1 2α−1 21−αln 2, z∈U. 2.69

Theorem 2.14. Leth∈ HU, withh0 1,h0/0 which satisfy the inequality

Re

1zhz

hz

>−1

2, z∈U. 2.70

Ifn, m∈N0, λ2≥λ1≥0,andf∈ Aand satisfies the diﬀerential subordination

μn,m_λ1,λ2fz ≺hz, z∈U, 2.71

then

μn,m_λ1,λ2fz z ≺qz

1

z

_z

0

htdt. 2.72

Proof. Let

pz μ

n,m λ1,λ2fz

z

_∞

k2

z

1p1zp2z2· · ·,

p∈ H1,1, z∈U.

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Diﬀerentiating2.73, with respect toz, we have

Using2.74, the diﬀerential subordination2.71becomes

pz zpz≺hz, z∈U. 2.75

FromLemma 1.1, we deduce that

pz≺qz 1 z

_z

0

htdt, 2.76

and using2.73, we obtain

μn,m_λ1,λ2fz z ≺qz

1

z

_z

0

htdt. 2.77

FromLemma 1.3, we have that the functionqis convex, and fromLemma 1.1,qis the best

dominant for subordination2.71. This completes the proof ofTheorem 2.14.

3. Conclusion

We remark that several subclasses of analytic univalent functions can be derived and studied using the operatorμn,m_λ1,λ2.

Acknowledgment

This work is fully supported by UKM-ST-06-FRGS0107-2009, MOHE, Malaysia.

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