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Volume 2009, Article ID 158408, 13 pagesdoi:10.1155/2009/158408 Research Article A New General Integral Operator Defined by Al-Oboudi Differential Operator Serap Bulut Civil Aviation Col

Volume 2009, Article ID 158408, 13 pages

doi:10.1155/2009/158408

Research Article

A New General Integral Operator Defined by

Al-Oboudi Differential Operator

Serap Bulut

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 ˙Izmit-Kocaeli, Turkey

Correspondence should be addressed to Serap Bulut,serap.bulut@kocaeli.edu.tr

Received 8 December 2008; Accepted 22 January 2009

Recommended by Narendra Kumar Govil

We define a new general integral operator using Al-Oboudi differential operator Also we introduce new subclasses of analytic functions Our results generalize the results of Breaz, G ¨uney, and S˘al˘agean

Copyrightq 2009 Serap Bulut 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

LetA denote the class of functions of the form

n2

which are analytic in the open unit diskU  {z ∈ C : |z| < 1}, and S : {f ∈ A : f is univalent

inU}

Iff is given by 1.1, then from 1.3 and 1.4 we see that

n2

1  n − 1λ k a n z n , k ∈ N0: N ∪ {0}, 1.5

Now we introduce new classesSk δ, b, λ and K k δ, b, λ as follows.

A functionf ∈ A is in the classes S k δ, b, λ, where δ ∈ 0, 1, b ∈ C−{0}, λ ≥ 0, k ∈ N0,

if and only if

Re



11

b





or equivalently

Re



1λ

b





for allz ∈ U.

A functionf ∈ A is in the classs K k δ, b, λ, where δ ∈ 0, 1, b ∈ C − {0}, λ ≥ 0, k ∈ N0,

if and only if

Re



1λ b z







for allz ∈ U.

We note thatf ∈ K k δ, b, λ if and only if zf

∈ Sk δ, b, λ.

S0δ, b, 1 ≡ S∗

introduced by Frasin3

ii For b  1 and λ  1, we have the class

iii In particular, the classes

are the classes of starlike functions of orderδ and convex functions of order δ in U,

resp-ectively

iv Furthermore, the classes

are familiar classes of starlike and convex functions inU, respectively

v For λ  1, we get

Let us introduce the new subclassesUSk α, δ, b, λ, UK k α, δ, b, λ and SH k α, b, λ,

KHk α, b, λ as follows.

A functionf ∈ A is in the class US k α, δ, b, λ if and only if f satisfies

Re



11b





> α

1

b





or equivalently

Re



1λ b





> α

λ b





whereα ≥ 0, δ ∈ −1, 1, α  δ ≥ 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0

A functionf ∈ A is in the class UK k α, δ, b, λ if and only if f satisfies

Re



1λ b z







> α

λ b



whereα ≥ 0, δ ∈ −1, 1, α  δ ≥ 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0

We note thatf ∈ UK k α, δ, b, λ if and only if zf

∈ USk α, δ, b, λ.

USk 0, δ, b, λ ≡ S k δ, b, λ, UK k 0, δ, b, λ ≡ K k δ, b, λ. 1.17

ii For b  1 and λ  1, we have the class

iii For λ  1, we have

iv For b  1 and λ  1, we have

Geometric Interpretation

and 1 λ/bzD k fz/D k fz, respectively, take all the values in the conic domain

R α,δ which is included in the right-half plane such that

From elementary computations we see that ∂R α,δ represents the conic sections symmetric about the real axis ThusR α,δ is an elliptic domain forα > 1, a parabolic domain

A functionf ∈ A is in the class SH k α, b, λ if and only if f satisfies

1

1

b





− 2α√2− 1

<Re



√ 2



11b





 2α√2− 1 z ∈ U,

1.22

whereα > 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0

A functionf ∈ A is in the class KH k α, b, λ if and only if f satisfies

1

λ b



√

2− 1

<Re



√ 2



1λ b z







 2α√2− 1 z ∈ U,

1.23

whereα > 0, b ∈ C − {0}, λ ≥ 0, k ∈ N0

We note thatf ∈ KH k α, b, λ if and only if zf

∈ SHk α, b, λ.

SHk α, 1, 1 ≡ SH k α,

defined in5

ii For λ  1, we have

D Breaz and N Breaz6 introduced and studied the integral operator

F n z  z

0



t

μ1

· · ·



t

μ n

wheref i ∈ A and μ i > 0 for all i ∈ {1, , n}.

By using the Al-Oboudi differential operator, we introduce the following integral operator So we generalize the integral operatorF n

0, and μ i > 0, 1 ≤ i ≤ n One defines the integral

I k,n,l,μ

 F,

0



t

μ1

· · ·



t

μ n

wheref1, , f n ∈ A and D is the Al-Oboudi differential operator.

i λ  1, then we have 7, Definition 1

ii λ  1, k  0 and l1  · · ·  l n  0, then we have the integral operator defined

by1.26

iii k  0, l1 · · ·  l n  l ∈ N0, then we have8, Definition 1.1

2 Main Results

The following lemma will be required in our investigation

Lemma 2.1 For the integral operator I k,n,l,μ f1, , f n   F, defined by 1.27, one has



n



i1

n



i1





z

μ1

· · ·



z

μ n

Also, using1.3 and 1.4, we obtain



On the other hand, from2.2 and 2.3, we find



i1



z

μ i

n

j1

j / i



z

μ j



Thus by2.2 and 2.4, we can write





n



i1





n

i1



D l i1f i z − D l i f i z



.

2.6

Finally, we obtain



n



i1





which is the desired result

Theorem 2.2 Let α i ≥ 0, δ i ∈ −1, 1, α i  δ i ≥ 0 1 ≤ i ≤ n, and b ∈ C − {0}, λ ≥ 0 Also suppose

that

n



i1

i1

Re



11b





for allz ∈ U By 2.1, we get

11b λz





D k Fz  1 

n



i1

b





 1 n

i1



11b





−n

i1

2.11

So,2.10 and 2.11 give us

Re



11b λz







 1 −n

i1

i1



1 1b





i1

i1

i 1  1 −

n



i1

1 α i

2.12

for allz ∈ U Hence, we obtain F ∈ K k γ, b, λ, where γ  1 −n i1 μ i 1 − δ i /1  α i

Corollary 2.3 Let α i ≥ 0, δ i ∈ −1, 1, α i  δ i ≥ 01 ≤ i ≤ n, and b ∈ C − {0} Also suppose that

n



i1

FromCorollary 2.3, we immediately getCorollary 2.4

Corollary 2.4 Let α i ≥ 0, δ i ∈ −1, 1, α i  δ i ≥ 0 1 ≤ i ≤ n, and b ∈ C − {0} Also suppose that

n



i1

an extension of Theorem 1

Corollary 2.6 Let δ i ∈ 0, 11 ≤ i ≤ n and b ∈ C − {0}, λ ≥ 0 Also suppose that

n



i1

If f i∈ Sl i δ i , b, λ1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the class

Kk ρ, b, λ, where

i1

Corollary 2.7 Let δ i ∈ 0, 1 1 ≤ i ≤ n and b ∈ C − {0} Also suppose that

n



i1

Sk1 ρ, b, 1, where ρ is defined as in 2.16.

Corollary 2.8readily follows fromCorollary 2.7

Corollary 2.8 Let δ i ∈ 0, 1 1 ≤ i ≤ n, and b ∈ C − {0} Also suppose that

n



i1

Sk1 0, b, 1.

Theorem 2.10 Let α i ≥ 0, δ i ∈ −1, 1, α i  δ i ≥ 0 1 ≤ i ≤ n and b ∈ C − {0}, λ ≥ 0 Also suppose

that

n



i1

Corollary 2.11 Let α i ≥ 0, δ i ∈ −1, 1, α i  δ i ≥ 0 1 ≤ i ≤ n and b ∈ C − {0} Also suppose that

n



i1

If f i∈ USl i α i , δ i , b, 1 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

Corollary 2.13 Let δ i ∈ 0, 11 ≤ i ≤ n and b ∈ C − {0}, λ ≥ 0 Also suppose that

n



i1

Kk ρ, b, λ, where ρ is defined as in 2.16.

Corollary 2.14 Let δ i ∈ 0, 11 ≤ i ≤ n and b ∈ C − {0} Also suppose that

n



i1

Sk1 ρ, b, 1, where ρ is defined as in 2.16.

Theorem 2.16 Let α ≥ 0, δ ∈ −1, 1, α  δ ≥ 0 and b ∈ C − {0}, λ ≥ 0 Also suppose that

n



i1

If f i ∈ USl i α, δ, b, λ 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

Re



11b





> α

1

b





for allz ∈ U.

On the other hand, from2.1, we obtain

1λ b z





D k Fz  1 

n



i1

b





 1 −n

i1

i1



11b





.

2.25

Considering1.16 with the above equality, we find

Re



1 λ b z







− α

λ b



− δ

 1 −n

i1

i1



11b





− α

n



i1

μ i b1





− δ

≥ 1 −n

i1

i1



11b





− αn

i1

1

b





− δ

i1

i1



1

b





 δ



− αn

i1

1

b





− δ

 1 − δ



1−n

i1



≥ 0

2.26 for allz ∈ U This completes proof.

Corollary 2.17 Let α ≥ 0, δ ∈ −1, 1, α  δ ≥ 0, and b ∈ C − {0} Also suppose that

n



i1

If f i ∈ USl i α, δ, b, 1 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

Theorem 2.19 Let α ≥ 0, b ∈ C − {0}, and λ ≥ 0 Also suppose that

n



i1

If f i ∈ SHl i α, b, λ 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

Proof Since f i∈ SHl i α, b, λ 1 ≤ i ≤ n, by 1.22 we have

Re



√

2

√

2

b





 2α√2− 1−

1

1

b





− 2α√2− 1

>0

2.29 for allz ∈ U Considering this inequality and 2.1, we obtain

Re

√

2

√

2

b





 2α√2− 1−

1

1

b



√

2− 1

 Re

√

2

√ 2

b

n



i1





 2α√2− 1

−

1

1

b

n



i1





− 2α√2− 1

√2n

i1

√

2

b





 2α√2− 1

−

1

n



i1

b





− 2α√2− 1

√2n

i1



√

2

√ 2

b





−√2

n



i1

−

1

n



i1

 11b





−2α√2−1



−n

i1

i1

√2



1−n

i1



 2α√2− 1n

i1



√

2

√ 2

b





−



1− 2α√2− 1



1−n

i1



n

i1



11b





− 2α√2− 1

≥√2



1−n

i1



 2α√2− 1n

i1



√

2

√ 2

b





− 1− 2α√

2− 1 1−n

i1



−n

i1

μ i

1

1

b





− 2α√2− 1

 2α√2− 1n

i1

i1

√2 2α√2− 1− 1− 2α√

2− 1 1−n

i1



n

i1



Re



√

2

√ 2

b





2α√2−1−

1

1

b





−2α√2−1



2− 1 1−n

i1



>



1−n

i1



min√

2− 11  4α,√2 1≥ 0

2.30

for allz ∈ U Hence by 1.23, we have F ∈ KH k α, b, λ.

Corollary 2.20 Let α ≥ 0 and b ∈ C − {0} Also suppose that

n



i1

If f i ∈ SHl i α, b, 1 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

Theorem 2.22 Let α ≥ 0, b ∈ C − {0} and λ ≥ 0 Also suppose that



1√2α√2− 1n

i1

If f i ∈ SHl i α, b, λ 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

Re



√

2

√

2

b





 2α√2− 1>

1

1

b





− 2α√2− 1

2.33

for allz ∈ U Considering this inequality and 2.1, we obtain

√

2Re



1λ b z







 Re



√

2

√ 2

b

n



i1





√2−√2

n



i1

i1



√

2

√ 2

b





√2−√2

n



i1

i1

i1

 Re



√ 2

√ 2

b





2α√2−1





1−1√2α√2− 1n

i1



> 0

2.34 for allz ∈ U Hence, by 1.8, we have F ∈ K k 0, b, λ.

Corollary 2.23 Let α ≥ 0 and b ∈ C − {0} Also suppose that



1√2α√2− 1n

i1

If f i ∈ SHl i α, b, 1 1 ≤ i ≤ n, then the integral operator I k,n,l,μ  F, defined by 1.27, is in the

References

1 F M Al-Oboudi, “On univalent functions defined by a generalized S˘al˘agean operator,” International Journal of Mathematics and Mathematical Sciences, vol 2004, no 27, pp 1429–1436, 2004.

2 G S¸ S˘al˘agean, “Subclasses of univalent functions,” in Complex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol 1013 of Lecture Notes in Mathematics, pp 362–372, Springer, Berlin,

Germany, 1983

3 B A Frasin, “Family of analytic functions of complex order,” Acta Mathematica Academiae Paedagogicae Ny´ ıregyh´aziensis, vol 22, no 2, pp 179–191, 2006.

4 I Magdas¸, Doctoral thesis, University “Babes¸-Bolyai”, Cluj-Napoca, Romania, 1999

5 M Acu, “Subclasses of convex functions associated with some hyperbola,” Acta Universitatis Apulensis,

no 12, pp 3–12, 2006

6 D Breaz and N Breaz, “Two integral operators,” Studia Universitatis Babes¸-Bolyai Mathematica, vol 47,

no 3, pp 13–19, 2002

7 D Breaz, H ¨O G ¨uney, and G S¸ S˘al˘agean, “A new general integral operator,” Tamsui Oxford Journal of Mathematical Sciences Accepted.

8 S Bulut, “Some properties for an integral operator defined by Al-Oboudi differential operator,” Journal

of Inequalities in Pure and Applied Mathematics, vol 9, no 4, article 115, 2008.