Vol. 17, No. 1 (2011), pp. 29–38.
ON CLASSES OF ANALYTIC FUNCTIONS
CONTAINING GENERALIZATION OF INTEGRAL
OPERATOR
Maslina Darus
1,∗and Rabha W. Ibrahim
21School of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia Bangi 43600, Selangor Darul Ehsan, Malaysia, [email protected]
2School of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia Bangi 43600, Selangor Darul Ehsan,
Malaysia, [email protected]
Abstract. New classes containing generalization of integral operator are intro-duced. Characterization and other properties of these classes are investigated. Fur-ther, Fekete-Szeg¨o functional for these classes are also given.
Key words and Phrases: Integral operator, analytic functions, differential operator, Fekete-Szeg¨o functional, distortion theorem.
Abstrak.Pada paper ini diperkenalkan kelas baru yang memuat perumuman dari operator integral. Kemudian karakterisasi dan sifat lain juga dikaji dari kelas. Lebih lanjut, fungsional Fekete-Szeg¨o untuk kelas ini juga diberikan.
Kata kunci: Perumuman operator integral, fungsi analitik, operator diferensial, fungsional Fekete-Szeg¨o, teorema distorsi.
1. Introduction
LetHbe the class of functions analytic in U andH[a, n] be the subclass of Hconsisting of functions of the formf(z) =a+anzn+an+1zn+1+... .LetAbe the subclass ofHconsisting of functions of the form
f(z) =z+
∞
X
n=2
anzn, z ∈U. (1)
2000 Mathematics Subject Classification: 30C 45.
Received: 17-12-2010, revised: 07-04-2011, accepted: 09-04-2011. *Corresponding author
And letFk(−1)be defined such that
Note that Fk(−1) is needed to get the integral operator from differential operator.
Then
k= 0 gives Noor integral operator [15,16].
Some of relations for this integral operator are discussed in the next lemma.
Lemma 1.1. Let f ∈ A.Then
Note that (ii) is the type of Bernardi integral[21].
Definition 1.1. Let f(z) ∈ A. Thenf(z) ∈Sk,α,βλ,δ (µ)if and only if
sharp upper bound of |a2| and of the Fekete-Szeg¨o functional |a3−νa22| for the
2. General Properties of Jk,α,βλ,δ
In this section we study the characterization properties and distortion theo-rems for the functionf(z) ∈ Ato belong to the classesSλ,δk,α,β(µ) andCk,α,βλ,δ (µ) by obtaining the coefficient bounds.
Theorem 2.1. Letf(z) ∈ A. If forα≥0, β≥1, δ≥0 andλ≥0
Proof. Suppose that (4) holds. Since
1−µ≥
then this implies that
1 +P∞n=2[(βn)α+(n−n1)(βn)|an| αλ]kC(δ,n)
We also note that the assertion (4) is sharp and the extremal function is given by
In the same way we can verify the following results
Theorem 2.2. Letf(z) ∈ A. If forα≥0, β≥1, δ≥0 andλ≥0
∞
X
n=2
n(n−µ)|an|
[(βn)α+ (n−1)(βn)αλ]kC(δ, n) ≤1−µ, 0≤µ <1, (5)
thenf(z) ∈Ck,α,βλ,δ (µ).The result (8) is sharp.
Theorem 2.3. Let the hypotheses of Theorem 2.1 be satisfy. Then forz ∈U and 0≤µ <1
|Jk,α,βλ,δ f(z)| ≥ |z| −1−µ 2−µ
and
|Jk,α,βλ,δ f(z)| ≤ |z|+1−µ 2−µ.
Proof. By using Theorem 2.1, one can verify that
(2−µ)
∞
X
n=2
|an|
[(βn)α+ (n−1)(βn)αλ]kC(δ, n)
≤
∞
X
n=2
(n−µ)|an|
[(βn)α+ (n−1)(βn)αλ]kC(δ, n)
≤1−µ
then
∞
X
n=2
|an|
[(βn)α+ (n−1)(βn)αλ]kC(δ, n) ≤ 1−µ
2−µ.
Thus we obtain
|Jk,α,βλ,δ f(z)| ≤ |z|+
∞
X
n=2
|an|
[(βn)α+ (n−1)(βn)αλ]kC(δ, n)|z| n
≤ |z|+
∞
X
n=2
|an|
[(βn)α+ (n−1)(βn)αλ]kC(δ, n)|z| 2
≤ |z|+ [1−µ 2−µ]|z|
The other assertion can be proved as follows
This complete the proof.
Theorem 2.4.Let the hypotheses of Theorem 2.21 be satisfy. Then forz ∈U and 0≤µ <1
Theorem 2.5. Let the hypotheses of Theorem 2.1 be satisfy. Then
(n−µ)
Proof. In virtue of Theorem 2.1, we have
Thus we obtain
|f(z)|=|z+
∞
X
n=2 anzn|
≤ |z|+
∞
X
n=2
|an||z|2
≤ |z|+ (1−µ)|z|2 The other assertion can be proved as follows
|f(z)| ≥ |z−
∞
X
n=2 anzn|
≥ |z| −
∞
X
n=2
|an||z|2
≥ |z| −(1−µ)|z|2.
This complete the proof.
Theorem 2.6. Let the hypotheses of Theorem 2.2 be satisfy. Then (n−µ)
[(βn)α+ (n−1)(βn)αλ]kC(δ, n) ≥1,∀n≥2 and0≤µ <1 poses
|f(z)| ≥ |z| − (1−µ) 2 |z|
2
and
|f(z)| ≤ |z|+(1−µ) 2 |z|
2 .
3. Fekete-Szeg¨o for the Classes Sk,α,βλ,δ (µ)andCk,α,βλ,δ (µ)
In this section we determine the sharp upper bound for |a2| for the classes
Sk,α,βλ,δ (µ) andCk,α,βλ,δ (µ).Moreover, we calculate the Fekete-Szeg¨o|a3−νa22| func-tional for them.
Theorem 3.1. Let the hypotheses of Theorem 2.1 be satisfy. Then
|a2| ≤
2(1−µ)[(β2)α(1 +λ)]kC(δ,2) 1 +µ
and for allν ∈Cthe following bound is sharp
|a3−νa22| ≤2(1−µ)[(3β)α(1 + 2λ)]kC(δ,3)
maxn1,|1 + 2(1−µ)
(1 +µ)[(3β)α(1 + 2λ)]kC(δ,3)
−2ν 1−µ
(1 +µ)2
{[(2β)α(1 +λ)]kC(δ,2)}2 [(3β)α(1 + 2λ)]kC(δ,3) |
o
Proof. Sincef ∈Sk,α,βλ,δ (µ) then the condition
ℜnz[J
k,α,β λ,δ f(z)]′
Jk,α,βλ,δ f(z)
o
> µ, 0≤µ <1, z ∈U
is equivalent to
z[Jk,α,βλ,δ f(z)]′= (1−µ)p(z)Jk,α,βλ,δ f(z), z ∈U,
for somep ∈ P.Equating coefficients we obtaina2=(1−1+µµ)Ap1, a3= (1−µ)B(p2+ (1−µ)
2+µ p 2
1) where A := [(2β)α(1 +λ)]kC(δ,2), B := [(3β)α(1 + 2λ)]kC(δ,3) and further, for C:= (11+µ−µ)2 +(1−2µ)B −ν((11+µ−µ)A)2 and by using Lemma 1.2 we have
|a3−νa22| ≤ H(x) = 2(1−µ)B+ (C− (1−µ)B
2 )x
2, x := |p
1|. Consequently, we receive
|a3−νa22|=
H(0) = 2(1−µ)B, C ≤(1−2µ)B
H(2) = 4C, C > (1−2µ)B.
Equality is attained for functions given by
z[Jk,α,βλ,δ f(z)]′
Jk,α,βλ,δ f(z) =
1 +z2(1−2µ) 1−z2
and
z[Jk,α,βλ,δ f(z)]′
Jk,α,βλ,δ f(z) =
1 +z(1−2µ) 1−z
respectively.
Forµ= 0 we receive the following corollary.
Corollary 3.1. Let the assumptions of Theorem 3.1 hold. Then forµ= 0
|a2| ≤2[(2β)α(1 +λ)]kC(δ,2) and
|a3−νa22| ≤2[(3β)α(1 + 2λ)]kC(δ,3)
maxn1,|1 + 2
[(3β)α(1 + 2λ)]kC(δ,3)
−2ν{[(2β)
α(1 +λ)]kC(δ,2)}2 [(3β)α(1 + 2λ)]kC(δ,3) |
o
.
In the similar manner we can prove the following result.
Theorem 3.2. Let the hypotheses of Theorem 2.2 be satisfy. then
|a2| ≤
and for allν ∈Cthe following bound is sharp
|a3−νa22| ≤
2(1−µ) 3(2 +µ)[(3β)
α(1 + 2λ)]kC(δ,3)
maxn1,|1 2+
2(1−µ) (3 +µ)
−3ν(1−µ) (3 +µ)2 {[(2β)
α(1 +λ)]kC(δ,2)}2|o .
Forµ= 0 we receive the following corollary.
Corollary 3.2. Let the assumptions of Theorem 3.2 hold. Then forµ= 0
|a2| ≤ 2 3[(2β)
α(1 +λ)]kC(δ,2)
and
|a3−νa22| ≤ 1 3[(3β)
α(1+2λ)]kC(δ,3)maxn1,|1 2+
2 3−
ν
3{[(2β)
α(1+λ)]kC(δ,2)}2|o.
Acknowledgement: The work here is fully supported by MOHE: UKM-ST-06-FRGS0244-2010.
References
[1] Al-Oboudi, F., ”On univalent functions defined by a generalized Salagean operator”, I.J.M.M.S,27(2004), 1429–1436.
[2] Al-Shaqsi, K. and Darus, M., ”An operator defined by convolution involving polylogarithms functions”,Journal of Mathematics and Statistics,4(2008), 46–50.
[3] Darus, M. and Ibrahim, R.W., ”Generalization of differential operator”,Journal of Mathe-matics and Statistics,4(2008), 138–144.
[4] Darus, M. and Ibrahim, R.W., ”On subclasses of uniformly Bazilevic type functions involving generalised differential and integral operators”,Far East Journal of Mathematical Sciences (FJMS),33(2009), 401–411.
[5] Darus, M. and Ibrahim, R.W., ”On subclasses for generalized operators of complex order”, Far East Journal of Mathematical Sciences (FJMS),33(2009), 299–308.
[6] Darus, M. and Ibrahim, R.W., ”On inclusion properties of generalized integral operator involving Noor integral”, Far East Journal of Mathematical Sciences (FJMS),33(2009), 309–321.
[7] Darus, M. and Ibrahim, R.W., ”Ibrahim and Iqbal H. Jebril, Bounded turning for generalized integral operator”,Int. J. Open Problems Complex Analysis,1( 2009), 1–10.
[8] Darus, M. and Ibrahim, R.W., ”New Classes Containing Generalization of Differential Op-erator”,Applied Mathematical Sciences,3:51(2009), 2507–2515.
[9] Darus, M. and Ibrahim, R.W., ”Integral operator defined byk−th Hadamard product”,ITB J. Sci.,42:2(2010), 135–152.
[11] Darus, M. and Ibrahim, R.W., ”Multivalent functions based on linear integral operator”, Miskolc N.N,11:1(2010), 43–52.
[12] Darus, M. and Ibrahim, R.W., ”Operator defined by convolution with Zeta functions”,Far East Journal of Mathematical Sciences,40:1(2010), 93–105.
[13] Darus, M. and Ibrahim, R.W., ”On univalent functions defined by a generalized differential operator”,J. Anal. Appl.,16:2(2010), 305–313.
[14] Duren, P.L.,Univalent functions, Springer-Verlag, New York, 1983.
[15] Noor, K.I., ”On new classes of integral operators”,J. Nat. Geomet.,16(1999), 71–80. [16] Noor, K.I. and Noor, M.A., ”On integral operators”, J. of Math. Anal. Appl.,238(1999),
341–352.
[17] Ruscheweyh, S., ”New criteria for univalent functions”,Proc. Amer. Math. Soc.,49(1975), 109–115.
[18] Sˇalˇagean, G. S., ”Subclasses of univalent functions”,Lecture Notes in Math., 1013, Springer-Verlag, Berlin, 1983, 362–372.
[19] Srivastava, H. M., Darus, M., and Ibrahim, R.W., ”Classes of analytic functions with frac-tional powers defined by means of a certain linear operator”,Integ. Tranc. Special Funct., 22(2011), 17–28.
[20] Darus M. and Al-Shaqsi K., ”Differential sandwich theorem with generalised derivative op-erator”,Int. J. Comput. Math. Sci.,2:2(2008), 75–78.