Abstract
In this article, we define and investigate new families of certain subclasses of meromorphic functions of complex order. Considering the new subclasses, several properties for certain integral operators are derived.
2010 Mathematics Subject Classification: 30C45.
Keywords:
Analytic function; meromorphic function; integral operator1 Introduction
Let Σ denote the class of functions of the form:
which are analytic in the punctured open unit disk
We say that a function f ∈ Σ is meromorphic starlike of order δ (0 ≤ δ < 1), and belongs to the class Σ*(δ), if it satisfies the inequality:
A function f ∈ Σ is a meromorphic convex function of order δ (0 ≤ δ < 1), if f satisfies the following inequality:
and we denote this class by Σ_{k}(δ).
For f ∈ Σ, Wang et al. [1] and Nehari and Netanyahu [2] introduced and studied the subclass Σ_{N}(β) of Σ consisting of functions f(z) satisfying
Let denote the class of functions f of the form , which are analytic in the open unit disk .
Analogous to several subclasses [310] of analytic functions of , we define the following subclasses of Σ.
Definition 1.1 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
Definition 1.2 Let a function f ∈ Σ be analytic in . Then f is in the class ΣK_{b}(δ) if, and only if, f satisfies
where b ∈ C\{0}, 0 ≤ δ < 1. We note that f ∈ Σ K_{b}(δ) if, and only if, .
Furthermore, the classes
are the classes of meromorphic starlike functions of order δ and meromorphic convex functions of order δ in , respectively. Moreover, the classes
are familiar classes of starlike and convex functions in , respectively.
Definition 1.3 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α ≥ 0, δ ∈ [1,1), α + δ ≥ 0, .
Definition 1.4 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α ≥ 0, δ ∈ [1,1), α + δ ≥ 0, .
We note that if, and only if, .
For α = 0, we have
Definition 1.5 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
Definition 1.6 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
We note that if, and only if, .
Let us also introduce the following families of new subclasses , , and as follows.
Definition 1.7 Let a function f ∈ Σ be analytic in . Then f is in the class , if, and only if, f satisfies
We note that if, and only if, .
Definition 1.8 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α ≥ 0, δ ∈ [1,1), α + δ ≥ 0, .
We note that if, and only if, .
Definition 1.9 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
We note that if, and only if, .
Recently, many authors introduced and studied various integral operators of analytic and univalent functions in the open unit disk [1121].
Most recently, Mohammed and Darus [22,23] introduced the following two general integral operators of meromorphic functions Σ:
and
Goyal and Prajapat [24] obtained the following results for f ∈ Σ to be in the class Σ*(δ), 0 ≤ δ < 1.
Corollary 1.1 If f ∈ Σ satisfies the following inequality
then f ∈ Σ*(δ).
Corollary 1.2 If f ∈ Σ satisfies the following inequality
then f ∈ Σ*.
Corollary 1.3 If f ∈ Σ satisfies the following inequality
then f ∈ Σ*.
In this article, we derive several properties for the integral operators and of the subclasses given by (1.5) and Definitions 1.1 to 1.6.
2 Some properties for
In this section, we investigate some properties for the integral operator defined by (1.15) of the subclasses given by (1.5), Definitions 1.1, 1.3, and 1.5.
Theorem 2.1 For i ∈ {1,..., n}, let γ_{i }> 0, f_{i }∈ Σ and
If
Proof On successive differentiation of , which is defined in (1.5), we get
and
Then from (2.3) and (2.4), we obtain
By multiplying (2.5) with z yield
This is equivalent to
Therefore, we have
Then, we easily get
Taking real parts of both sides of (2.9), we obtain
Let
Since , applying Corollary 1.1, we have
Then, by the hypothesis (2.2), we have β > 1. Therefore, , where β > 1. This completes the proof. □
Letting δ = 0 in Theorem 2.1, we have
Corollary 2.2 For i ∈ {1,..., n}, let γ_{i }> 0, f_{i }∈ Σ and
If
Making use of (2.11), Corollary 1.2 and Corollary 1.3, one can prove the following results.
Theorem 2.3 For i ∈ {1,..., n}, let γ_{i }> 0, f_{i }∈ Σ and
If
Theorem 2.4 For i ∈ {1,..., n}, let γ_{i }> 0, f_{i }∈ Σ and
If
Theorem 2.5 For i ∈ {1,..., n}, let γ_{i }> 0 and (0 ≤ δ < 1 and ).
If
Proof From (2.7), we have
Equivalently, (2.20) can be written as
Taking the real part of both terms of the last expression, we have
so that
Now, adopting the techniques used by Breaz et al. [11] and Bulut [21], we prove the following two theorems.
Theorem 2.6 For i ∈ {1,..., n}, let γ_{i }> 0 and (α ≥ 0, δ ∈ [1,1), α + δ ≥ 0 and ). If
Proof Since , it follows from Definition 1.3 that
Considering Definition 1.8 and with the help of (2.21), we obtain
This completes the proof. □
Theorem 2.7 For i ∈ {1,..., n}, let γ_{i }> 0 and (α > 0 and ). If
Proof Since , it follows from Definition 1.5 that
Considering this inequality and (2.21), we obtain
This completes the proof. □
3 Some properties for
In this section, we investigate some properties for the integral operator defined by (1.16) of subclasses given by (1.5), Definitions 1.2, 1.4, and 1.6.
Theorem 3.1 For i ∈ {1,..., n}, let γ_{i }> 0, f_{i }∈ Σ and
Proof On successive differentiation of , which is defined in (1.16), we have
and
Then from (3.2) and (3.3), we obtain
By multiplying (3.4) with z yields,
that is equivalent to
Therefore, we have
so that
Taking the real parts of both terms of the last expression, we obtain
Let
Since , f_{i }∈ Σ_{k}(δ), we get
which, in light of the hypothesis (3.1), yields β > 1.
Therefore, , β > 1. This completes the proof. □
Theorem 3.2 For i ∈ {1,..., n}, let γ_{i }> 0, f_{i }∈ Σ and
Proof It follows from (3.7) that
Taking the real parts of both terms of the last expression, we obtain
Thus, we have
Let
Then, by the hypothesis (3.12), we see that β > 1. Therefore, , β > 1. This completes the proof. □
Now, using the method given in the proofs of Theorems 2.5, 2.6, and 2.7, one can prove the following results:
Theorem 3.3 For i ∈ {1,..., n}, let γ_{i }> 0 f_{i }∈ ΣK_{b}(δ_{i}) (0 ≤ δ < 1 and )). If
Theorem 3.4 For i ∈ {1,..., n}, let γ_{i }> 0 and (α ≥ 0, δ ∈ [1,1), α + δ ≥ 0 and )). If
Theorem 3.5 For i ∈ {1,..., n}, let γ_{i }> 0 and (α ≥ 0, and )). If
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
AM is currently a PhD student under supervision of MD and jointly worked on deriving the results. All the authors read and approved the final manuscript.
Acknowledgements
The above study was supported by MOHE grant: UKMST06FRGS02442010.
References

Wang, ZG, Sun, Y, Zhang, ZH: Certain classes of meromorphic multivalent functions. Comput Math Appl. 58, 1408–1417 (2009). Publisher Full Text

Nehari, Z, Netanyahu, E: On the coefficients of meromorphic Schlicht functions. Proc Am Math Soc. 8, 15–23 (1957). Publisher Full Text

Frasin, BA: Family of analytic functions of complex order. Acta Math Acad Paed Ny. 22(2), 179–191 (2006)

Ronning, F: Uniformly convex functions. Ann Polon Math. 57, 165–175 (1992)

Murugusundaramoorthy, G, Maghesh, N: A new subclass of uniforlmly convex functions and a corresponding subclass of starlike functions with fixed second coefficient. J Inequal Pure Appl Math. 5(4), 1–10 (2005)

Stankiewicz, J, Wisniowska, A: Starlike functions associated with some hyperbola. Folia Scientiarum Universitatis Tehnicae Resoviensis 147, Matematyka. 19, 117–126 (1996)

Acu, M: Subclasses of convex functions associated with some hyperbola. Acta Universitatis Apulensis. 12, 3–12 (2006)

Acu, M, Owa, S: Convex functions associated with some hyperbola. J Approx Theory Appl. 1, 37–40 (2005)

Nasr, MA, Aouf, MK: Starlike function of complex order. J Nat Sci Math. 25(91), 1–12 (1985)

Baricz, Á, Frasin, BA: Univalence of integral operators involving Bessel functions. Appl Math Lett. 23, 371–376 (2010). Publisher Full Text

Mohammed, A, Darus, M: New properties for certain integral operators. Int J Math Anal. 4(42), 2101–2109 (2010)

Frasin, BA: Univalence of two general integral operators. Filomat. 23, 223–229 (2009). Publisher Full Text

Breaz, D, Breaz, N, Srivastava, HM: An extension of the univalent condition for a family of integral operators. Appl Math Lett. 22, 41–44 (2009). Publisher Full Text

Breaz, D, Breaz, N: Two integral operators. Studia Universitatis BabesBolyai, Mathematica, ClujNapoca. 3, 13–21 (2002)

Breaz, D, Owa, S, Breaz, N: A new integral univalent operator. Acta Univ Apulensis Math Inform. 16, 11–16 (2008)

Breaz, D, Güney, HÖ, Salagean, GS: A new general integral operator. Tamsui Oxford, J Math Sci. 25(4), 407–414 (2009)

Darus, M, Faisal, I: A study on Beckers univalence criteria. Abs Appl Anal. 2011, 13 (2011) Article, ID 759175

Breaz, N, Braez, D, Darus, M: Convexity properties for some general integral operators on uniformly analytic functions classes. Comput Math Appl. 60, 3105–3107 (2010). Publisher Full Text

Latha, S, Dileep, L: On a generalized integral operator. Int J Math Anal. 3(30), 1487–1491 (2009)

Bulut, S: A new general integral operator defined by AlOboudi differential operator. J Inequal Appl. 2009, 13 (2009) Article, ID 158408

Mohammed, A, Darus, M: A new integral operator for meromorphic functions. Acta Universitatis Apulensis. 24, 231–238 (2010)

Mohammed, A, Darus, M: Starlikeness properties of a new integral operator for meromorphic functions. J Appl Math. 2011, 8 (2011) Article, ID 804150

Goyal, SP, Prajapat, JK: A new class of meromorphic multivalent functions involving certain linear operator. Tamsui Oxford, J Math Sci. 25(2), 167–176 (2009)