Preserving Classes of Meromorphic Functions through Integral Operators

: We consider three new classes of meromorphic functions deﬁned by an extension of the Wanas operator and two integral operators, in order to study some preservation properties of the classes. The purpose of the paper is to ﬁnd the conditions such that, when we apply the integral operator J p , γ to some function from the new deﬁned classes Σ S np , q ( α , δ ) , respectively Σ S np , q ( α ) , we obtain also a function from the same class. We also deﬁne a new integral operator on the class of meromorphic functions, denoted by J p , γ , h , where h is a normalized analytic function on the unit disc. We study some basic properties of this operator. Then we look for the conditions which allow this operator to preserve a particular subclass of the classes mentioned above.


Introduction and Preliminaries
Many operators have been used since the beginning of the study of analytic functions. The most interesting of these are the differential and integral operators. Since the beginning of the 20th century, many mathematicians have worked on integral operators applied to classes of analytic functions, but papers on integral operators applied to classes of meromorphic functions are smaller in number. This is happening because there is a need of new integral operators on meromorphic functions.
The first author of the present paper started in 2010 to work on integral operators on meromorphic functions (see [1]). In the same period, new results regarding the same topic were published in papers such as [2][3][4] etc.
The literature on meromorphic functions is very large, but in the field of geometric theory of meromorphic functions there is still more to say. Recent results on this topic may be found in [5][6][7][8][9].
In this work we introduce a new integral operator on the class of meromorphic functions and we prove that it is well defined. We also introduce new classes of meromorphic functions, with the use of the Wanas operator, and we study some preserving properties of these classes.
Using the integral operator introduced in this paper, beautiful results can be obtained in terms of class conservation.
For an analytic function f we have where the (p, q)-bracket number, or twin-basic [k] p,q , is given by which is a natural generalization of the q-number, and we have For more details on the concepts of (p, q)-calculus, or q calculus (in the case when p = 1), see [12,13].
For p ∈ N * , n ∈ N and 0 < q < 1 we consider the extension of the Wanas operator for meromorphic functions, denoted by W n q : Σ p → Σ p , as We have the properties:
The second equivalence can be proved in the same way.
We have It is well known that the operator can be also written as Therefore, we have this meaning that J p,γ (W n q (g)) = W n q (J p,γ (g)). We get that W n q (J p,γ (g)) ∈ Σ * p (α, δ), therefore J p,γ (g) ∈ ΣS n p,q (α, δ).
If we consider, in the above theorem, the case that n = 0 we obtain: The result of Corollary 4 was also found in [1].
Proof. We omit the proof since it is similar to the proof of Theorem 3, except that we now use, instead of Corollary 1, Corollary 2 with β = 1.
Next we define the operator J p,γ,h . Let p ∈ N * , γ ∈ C with Re γ > p and h ∈ A. We define J p,γ,h : It is easy to see that for the h(z) = z we have J p,γ,h = J p,γ , where found in [1], was used in different papers.
If G = J p,γ,h (g) is defined by (7), then G ∈ Σ p with z p G(z) = 0, z ∈ U, and All powers in (7) are principal ones.

Proof.
We consider Theorem 1 with Using the above notations we show that the subordination is equivalent to by using the logarithmic differential, we get We have now Therefore, the subordination from the hypothesis of Theorem 1 is satisfied. Since all the other conditions from the hypothesis of Theorem 1 are met, we get from Theorem 1 that belongs to the class Σ p with z p G(z) = 0, z ∈ U, and Taking into account the fact that we have Φ(z) = h(z) z γ , by using the logarithmic this meaning that the inequality is equivalent to the inequality Therefore, the proof of the theorem is complete.
If we consider in Theorem 5 that h(z) = z, since the requirements on h are satisfied, we get: Then G = J p,γ (g) ∈ Σ p with z p G(z) = 0, z ∈ U, and Re The above corollary is a particular case of Corollary 2 from [1] (considering β = 1).

Proposition 2.
Let p ∈ N * , γ ∈ C with Re γ > p and h ∈ A with h(z) z · h (z) = 0. We denote by H the function H(z) = h(z) h (z) . Let g ∈ Σ p and G = J p,γ,h (g). Then we have the equality From (9), after differentiating, we obtain We use the notation p(z) = − zG (z) G(z) and we get: From (9) and (10) we obtain In the last equality we replace G and G , from (11), and we get: .
Using now the notations from the hypothesis we obtain that Let us denote W n q g = g 1 . Since W n q g ∈ Σ * p ⊂ Σ p and satisfies (13) it follows from Theorem 5 that G 1 = J p,γ,h (g 1 ) ∈ Σ p with z p G 1 (z) = 0, z ∈ U, and Re zG 1 (z) From (15) we have G 1 = J p,γ,h (g 1 ) = J p,γ,h (W n q g) = W n q G, therefore (16) is the same with z p W n q G(z) = 0, z ∈ U, and Re We also have g 1 ∈ Σ * p , this meaning that Re − zg 1 (z) g 1 (z) > 0.
Since G 1 = J p,γ,h (g 1 ) we get from Proposition 2 that Next, we prove that α(z) ∈ R, z ∈ U.
We have because, from (14), On the other hand, since we obtain, from (16), Re R(z) > 0. We have the functions α : U → R, R : U → C with Re R(z) > 0, z ∈ U and p ∈ H[p, n]. Therefore, since we get from Lemma 2 that Re p(z) > 0, z ∈ U.
Thus, Re − zG 1 (z) G 1 (z) > 0, z ∈ U, this meaning that G 1 = W n q G ∈ Σ * p , which is equivalent to G ∈ ΣS n p,q . meromorphic functions. We appealed to the Wanas operator because we noticed that it is a well-known operator in recent papers. It is shown that classes of starlike functions of the order α are obtained for specific values of n. Some interesting preserving problems concerning these classes are discussed in the theorems and corollaries. We have given the conditions for having the function J p,γ (g) (where J p,γ is a wellknown integral operator) in one of the classes ΣS n p,q (α, δ), respectively ΣS n p,q (α), when g is a function from the same class. It can be seen that these conditions are relatively simple.
Next, we have introduced a new integral operator on meromorphic functions, denoted by J p,γ,h , proved that it is well-defined and looked for the conditions which allow this operator to preserve the class ΣS n p,q . The preservation of ΣS n p,q -like classes, following the application of this operator, can be investigated in future works.
Examples were given as corollaries for particular cases of the function h. The new operator defined in this paper can be used to introduce other subclasses of meromorphic functions. Quantum calculus can be also associated for future studies and symmetry properties can be investigated.