ON UNIVALENCE CRITERIA FOR ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED DIFFERENTIAL OPERATOR
Maslina Darus and Rabha W. Ibrahim
AbstractIn this paper we obtain sufficient conditions for univalence of analytic functions defined by a generalized differential operator introduced by the authors in (Far East J. Math. Sci., 33(3),(2009), 299-308).
Keywords and phrases: Differential operator; Analytic functions; Univalent func- tions.
2000AMS Mathematics Subject Classification: 30C45.
∗-corresponding author
1. Introduction.
LetHbe the class of functions analytic in U :={z∈C:|z|<1}andH[a, n] be the subclass ofHconsisting of functions of the formf(z) =a+anzn+an+1zn+1+... . Let Abe the subclass ofH consisting of functions of the form
f(z) =z+
∞
X
n=2
anzn, (z ∈U). (1)
We will use the following operator which defined and studied by the authors (see [1]).
D0f(z) =f(z)
=z+
∞
X
n=2
anzn,
Dα,β,λ1 f(z) = [1−β(λ−α)]f(z) +β(λ−α)zf0(z)
=z+
∞
X
n=2
[β(n−1)(λ−α) + 1]anzn,
... Dkα,β,λf(z) =D1α,β,λ
Dα,β,λk−1 f(z)
=z+
∞
X
n=2
[β(n−1)(λ−α) + 1]kanzn
(2)
forβ >0, 0≤α < λ andk∈N0 =N∪ {0}with Dα,β,λk f(0) = 0.
Remark 1.1.
(i) Whenα = 0, β= 1,we receive Al-Oboudi0s differential operator (see [2]).
(ii) And when α = 0, β = 1 and λ= 1 we get Sˇalˇagean0s differential operator (see [3]).
Many differential operators studied by various authors can be seen in the literature (see for examples [4]-[8]).
Our considerations are based on the following results.
Lemma 1.1. (see [9]) Let f ∈A. If for allz∈U (1− |z|2)|zf00(z)
f0(z) | ≤1, (3)
then the function f is univalent in U.
Lemma 1.2. (see [10])Let f ∈A. If for all z∈U
|z2f0(z)
f2(z) −1|<1, (4)
then the function f is univalent in U.
Lemma 1.3. (see [11])Let µ be a real number, µ > 12 and f ∈A. If for allz∈U
|(1− |z|2µ)zf00(z)
f0(z) + 1−µ| ≤µ, (5)
then the function f is univalent in U.
Lemma 1.4. (see [12])If f(z)∈S (the class of univalent functions) and z
f(z) = 1 +
∞
X
n=1
bnzn, (6)
then ∞
X
n=1
(n−1)|bn|2 ≤1.
Lemma 1.5. (see [13])Let ν ∈C,<{ν} ≥0 and f ∈A. If for allz∈U (1− |z|2<(ν))
<(ν) |zf00(z)
f0(z) | ≤1, (7)
then function
Fν(z) = ν
Z z
0
uν−1f0(u)du1/ν
is univalent in U.
1. The Main Results
In this section, we establish the sufficient conditions to obtain a univalence for analytic functions involving the differential operator (2).
Theorem 2.1. Let f ∈A.If for all z∈U
∞
X
n=2
[β(n−1)(λ−α) + 1]k[n(2n−1)]|an| ≤1. (8)
Then Dkα,β,λf(z) is univalent in U.
Proof. Letf ∈A.Then for allz∈U we have
(1− |z|2)|z[Dkα,β,λf(z)]00
[Dα,β,λk f(z)]0 | ≤(1 +|z|2)|z[Dα,β,λk f(z)]00 [Dkα,β,λf(z)]0 |
≤ 2P∞
n=2[β(n−1)(λ−α) + 1]k[n(n−1)]|an| 1−P∞
n=2[β(n−1)(λ−α) + 1]kn|an|
the last inequality is less than 1 if the assertion (8) is hold. Thus in view of Lemma 1.1, Dkα,β,λf(z) is univalent in U.
Theorem 2.2. Let f ∈A.If for all z∈U
∞
X
n=2
[β(n−1)(λ−α) + 1]k|an| ≤ 1
√7. (9)
Then Dkα,β,λf(z) is univalent in U.
Proof. Letf ∈A. It sufficient to show that
|z2[Dα,β,λk f(z)]0 2[Dα,β,λk f(z)]2| ≤1.
|z2[Dkα,β,λf(z)]0 2[Dkα,β,λf(z)]2| ≤ 1 +P∞
n=2[β(n−1)(λ−α) + 1]kn|an| 2
1−2P∞
n=2[β(n−1)(λ−α) + 1]k|an| −(P∞
n=2[β(n−1)(λ−α) + 1]k|an|)2 the last inequality is less than 1 if the assertion (9) is hold. Thus in view of Lemma 1.2, Dkα,β,λf(z) is univalent in U.
Theorem 2.3. Let f ∈A.If for all z∈U
∞
X
n=2
n[2(n−1) + (2µ−1)][β(n−1)(λ−α) + 1]k|an| ≤2µ−1, µ > 1
2. (10) Then Dkα,β,λf(z) is univalent in U.
Proof. Letf ∈A.Then for allz∈U we have
|(1− |z|2µ)z[Dα,β,λk f(z)]00
[Dkα,β,λf(z)]0 + 1−µ| ≤(1 +|z|2)|z[Dkα,β,λf(z)]00
[Dα,β,λk f(z)]0 |+|1−µ|
≤ 2P∞
n=2[β(n−1)(λ−α) + 1]k[n(n−1)]|an| 1−P∞
n=2[β(n−1)(λ−α) + 1]kn|an| +|1−µ|
the last inequality is less thanµif the assertion (10) is hold. Thus in view of Lemma 1.3, Dkα,β,λf(z) is univalent in U.
As applications of Theorems 2.1, 2.2 and 2.3 we have the following result.
Theorem 2.4. Let f ∈A. If for allz∈U one of the inequalities (8-10) holds then
∞
X
n=1
(n−1)|bn|2 ≤1,
where
z
Dα,β,λk f(z) = 1 +
∞
X
n=1
bnzn.
Proof. Letf ∈A.Then in view of Theorems 2.1 , 2.2 or 2.3,Dα,β,λk f(z) is univalent in U.Hence by Lemma 1.4 we obtain the result.
Theorem 2.5. Let f ∈A.If for all z∈U
∞
X
n=2
n[2(n−1) +<(ν)][β(n−1)(λ−α) + 1]k|an| ≤ <(ν), <(ν)>0. (11)
Then
Gν(z) =
ν Z z
0
uν−1[Dkα,β,λf(u)]0du 1/ν
is univalent in U.
Proof. Letf ∈A.Then for allz∈U we have (1− |z|2<(ν))
<(ν) |z[Dα,β,λk f(z)]00
[Dkα,β,λf(z)]0 | ≤ (1 +|z|2<(ν))
<(ν) |z[Dα,β,λk f(z)]00 [Dkα,β,λf(z)]0 |
≤ 2
<(ν) P∞
n=2[β(n−1)(λ−α) + 1]k[n(n−1)]|an| 1−P∞
n=2[β(n−1)(λ−α) + 1]kn|an| the last inequality is less than 1 if the assertion (11) is hold. Thus in view of Lemma 1.5, Gν(z) is univalent inU.
Acknowledgement: The work here is supported by UKM-ST-06-FRGS0107-2009.
References
[1] M. Darus and R. W. Ibrahim, On subclasses for generalized operators of complex order, Far East J. Math. Sci., 33(3),(2009), 299-308.
[2] F. M. Al-Oboudi, On univalent functions defined by a generalized Sˇalˇagean operator, Inter. J. Math. Math. Sci.,(27)(2004), 1429-1436.
[3] G. S. Sˇalˇagean, Subclasses of univalent functions, Lecture Notes in Math., 1013, Springer-Verlag, Berlin, (1983), 362-372.
[4] M.H. Al-Abbadi and M. Darus,Differential subordination for new generalised derivative operator, Acta Universitatis Apulensis. No. 20/2009,(2009), 265-280.
[5] M. Darus and R. W. Ibrahim, On new classes of univalent harmonic func- tions defined by generalized differential operator, Acta Universitatis Apulensis. No.
18/2009,(2009), 61-69.
[6] A. Ibrahim, S. Owa, M. Darus and Y. Nakamura,Generalization of Salagean operator for certain analytic functions, Banach J. Math. Anal. 2(2) (2008),16-22.
[7] K.Al-Shaqsi and M.Darus, An operator defined by convolution involving the polylogarithms functions, Journal of Mathematics and Statistics, 4(1) (2008),46-50.
[8] M.H. Al-Abbadi and M. Darus, Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions, International Journal of Mathematics and Mathematical Sciences, Volume 2010, Article ID 369078, 15 pages, doi:10.1155/2010/369078.
[9] J. Becker, L¨ownersche Differential gleichung und quasi-konform fortsetzbare schlichte funktionen , J. Reine Angew. Math., 255,(1972), 23-43.
[10] S. Ozaki, M. Nunokawa,The Schwarzian derivative and univalent functions, Proc. Amer. Math. Soc. 33(2), (1972), 392-394.
[11] H. Tudor,A sufficient condition for univalence, General Math., Vol. 17, No.
1 (2009), 89-94.
[12] A. W. Goodman, Univalent Functions, Vol.I, and II, Mariner, Tampa, Florida, 1983.
[13] N. N. Pascu , On the univalence criterion of Becker, Mathematica, Cluj- Napoca, Tome 29(52) Nr.2 (1987), 175-176.
Maslina Darus and Rabha W. Ibrahim School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia
Bangi 43600 Selangor D. Ehsan, Malaysia
email: [email protected], [email protected]
