On Classes of Meromorphic Functions Deﬁned by Subordination and Convolution

: For p ∈ N ∗ , let Σ p denote the class of meromorphic p-valent functions. We consider an operator for meromorphic functions denoted by T nb , which generalizes some previously studied operators. We introduce some new subclasses of the class Σ p , associated with subordination using the above operator, and we prove that these classes are preserved regarding the operator J p , γ , so we have symmetry when we look at the form of the class in which we consider the function g and at the form of the class of the image J p , γ ( g ) , where J p , γ ( g )( z ) = γ − p z γ (cid:90) z 0 g ( t ) t γ − 1 dt , γ ∈ C with Re γ > p .

For p ∈ N * , let Σ p denote the class of meromorphic functions of the form To introduce the next meromorphic function subclasses, we need to know what is a subordination.

Definition 1 ([1] (p. 4)).
Let f and F be members of H(U).The function f is said to be subordinate to F, written as f ≺ F or f (z) ≺ F(z), if there exists a function w analytic in U, with w(0) = 0 and |w(z)| < 1, and such that f (z) = F(w(z)).
ΣC p,0 (h 1 ; h) = g ∈ Σ p,0 : there exists ϕ ∈ ΣK p,0 (h) such that g (z) Inspired by the convolution of two analytic functions f and g, of the form We will denote by SC(U) It is obvious that every sequence satisfying the condition lim For different choices of the sequence b, we obtain particular forms of the operator T n b , forms that were studied in many previous articles.
For instance, if we consider b k = k, then the sum is the well-known Sǎlǎgean differential operator of order n [5].
We have the following proprieties for ).We make the remark that the properties (1)-( 5), of the operator T n b , are also shared by the above-mentioned operators W n q (see [10]) and J n p,λ (see [6]).The advantages of studying the operator T n b is that it has a simple form and generalizes other operators defined on the class of meromorphic functions.
Next, using the above operator T n b , we define some new subclasses of the class Σ p , associated with subordination, such that, in some particular cases, these new subclasses are the well-known classes of meromorphic starlike, convex or close-to-convex functions that were (and still are) studied by many authors working in the field of geometric function theory.We mention here some of the first papers dealing with these special classes of meromorphic functions [17][18][19].
If we consider the function g = p, this meaning that g(z) = 1 z p belongs to the new defined classes ΣS n p,b (h), respectively, ΣK n p,b (h).
It is obvious that for the above classes, we have Since in Mathematics symmetry means that one shape is identical to the other shape when it is moved, rotated, or flipped, we will see that the majority of our results have symmetry when we look at the form of the class of function g and at the form of the image class of J p,γ (g), this meaning that for g in ΣS n p,b (h), satisfying some conditions, we obtain J p,γ (g) in ΣS n p,b (h).We also have this symmetry property for all previously defined classes.
In the field of geometric theory of analytic functions, since the beginning of the 20th century, many mathematicians studied different operators applied to classes of analytic functions and their properties, but papers on operators applied to classes of meromorphic functions suffer from a gap.Also, there is still more to say about the property of preserving meromorphic classes and the symmetry properties when we apply an operator.
From 2011, the second author of the present paper began to study integral operators on meromorphic multivalent functions and classes obtained by using the analytic solution of a Briot-Bouquet differential equation which was the best (a, n)-dominant for a Briot-Bouquet subordination.We mention that a Briot-Bouquet subordination has the form For a better understanding of the expression "the best dominant" we give the next definition: then p is called a solution for (1).The univalent function q is said to be a dominant if we have p ≺ q for all p verifying (1).A dominant q with q ≺ q for all dominants q of (1) is called the best dominant of (1).
Briot-Bouquet differential subordinations and their solutions began to be studied intensively by P. J. Eenigenburg, S. S. Miller, P. T. Mocanu and M. O. Reade in 1983 (see [20]).Very important and useful results regarding the Briot-Bouquet differential equations and subordinations were published some years later by S. S. Miller, P. T. Mocanu in [21].To obtain our result, we turned to the solutions of these so-called Briot-Bouquet differential subordinations.

Main Results
First of all, for p, n, m ∈ N, p = 0, n > m, we give a link between the sets The second equivalence can be proved in the same way.
From Proposition 1, considering m = n − 1, we have a link between the sets ΣS n p,b (h) and ΣS n−1 p,b (h), respectively, ΣK n p,b (h) and ΣK n−1 p,b (h), as we can see: Proof.From g ∈ ΣC n p,0,b (h 1 ; ϕ, h), we have The next result is a lemma that is needed to prove a theorem, which will help us to obtain functions from the class ΣC n p,0,b (h 1 ; ϕ, h) when a function from the class ΣK n p,0,b (h) is already given.
Proof.First of all, we make the remark that, since g ∈ Σ p,0 , we have g of the form and we find that From the definition of the function f λ,g , we have We notice that for λ = 0 in Lemma 1, we find that f 0,g = g ∈ ΣC p,0 (1; g, h) ⊂ ΣC p,0 (1; h), so we have the next result: Corollary 1.Let p ∈ N * , h ∈ H(U) with h(0) = p.We have Proof.Since g ∈ ΣK n p,0,b (h), we have from Definition 2 that T n b (g) ∈ ΣK p,0 (h), so, from Lemma 1 we obtain where On the other hand, from Proposition 3, we have By using the properties of the operator T n b , we see that We notice that for λ = 0 in Theorem 1 we obtain that so we have the next result: We consider now the integral operator J p,γ (see [4]), defined on the class of meromorphic function Σ p , where γ ∈ C with Re γ > p.We give the conditions such that J p,γ preserves the classes defined in Definition 2, respectively, in Definition 3.
where q is the univalent solution of the Briot-Bouquet differential equation The function q is the best (p,p)-dominant.
, Corolarry 2.13, since the hypotheses are verified, we find that where q is the univalent solution of the Briot-Bouquet differential equation and the function q is the best (p,p)-dominant.
We know from [6] that this meaning that By using the definition of the operator T n b , we obtain From ( 3) and ( 4), we obtain which, from Definition 2, is equivalent to where q is the univalent solution of the Briot-Bouquet differential equation and the function q is the best (p,p)-dominant.
Since q ≺ h, we have ΣS n p,b (q) ⊂ ΣS n p,b (h), so we obtain the next corollary: For the next theorem, we will omit the proof because it is similar to the previous one.For the proof of Theorem 3 we use [3], Corollary 2.3.
where q is the univalent solution of the Briot-Bouquet differential equation q(z) + (p + 1)zq (z) with q(0) = p.The function q is the best (p,p+1)-dominant.
Since for Theorem 3 we have q ≺ h, we obtain the next corollary: where q is the univalent solution of the Briot-Bouquet differential equation q(z) + (p + 1)zq (z) with q(0) = p.The function q is the best (p,p+1)-dominant.
It is easy to see from the proof of Theorem 4 that we also have the next result: Let h 1 and h be convex functions in U with h 1 (0) = 1, h(0) = p and Re h(z) < Re (γ), z ∈ U.If g ∈ ΣC n p,0,b (h 1 ; h), then J p,γ (g) ∈ ΣC n p,0 (h 1 ; q), where q is the univalent solution of the Briot-Bouquet differential equation q(z) + (p + 1)zq (z) with q(0) = p.The function q is the best (p,p+1)-dominant.
Using the previous Corollary for m times (m ∈ N * ), we find that we have The operator J m p,γ was previously used in the paper [6].The last result will present the class of the image of the function f λ,g , when we have g ∈ ΣK n p,0,b (h), through the operator J p,γ .

Conclusions
In this paper, we first introduce a new operator on the classes of meromorphic multivalent functions denoted by T n b using the well-known convolution.This operator, for different choices of the sequence b, becomes an operator that was previously studied, so the operator T n b is a generalization of some operators, which verify some special properties, studied before.The advantages of studying the operator T n b is that it has a simple form and generalizes other operators, sharing also the same basic properties.Then we build new classes of meromorphic functions, using the operator T n b and the subordination, denoted by ΣS n p,b (h), ΣK n p,b (h), ΣC n p,0,b (h 1 ; h).It is obvious that classes of starlike meromorphic functions, convex and close-to-convex meromorphic functions are obtained from the abovedefined classes when n = 0 and for a specific function h.Some interesting preserving properties, concerning these classes, are discussed in theorems and corollaries, when we apply the well-known integral operator J p,γ .To obtain our results, we also turned to the solutions of so-called Briot-Bouquet differential subordinations and to the best dominant for the considered subordination.We mention also here that a Briot-Bouquet differential subordination is a differential subordination that has the form p(z) + nzp (z) βp(z) + γ ≺ h(z), z ∈ U, p, h ∈ H(U), with p(0) = h(0).
We make the remark that ΣS n p,b (h)-like classes may be defined using also the superordination and the preservation of such a class, following the application of the operator J p,γ can be investigated in future works.
that is denoted by f * g and is the analytic function( f * g)(z) = z + ∞ ∑ k=2 a k b k z k ,we consider an extension of it on the class of meromorphic functions, thus defining a new operator denoted by T n b , as it follows: For p ∈ N * , n ∈ N and b = (b k ) k≥0 , a sequence of complex numbers, such that ∞ ∑ k=0 b k z k is convergent for z ∈ U, we consider the new operator for meromorphic functions, denoted by T n b : We have g ∈ ΣC n p,0,b (h 1 ; ϕ, h) ⇔ T n−m b (g) ∈ ΣC m p,0,b (h 1 ; T n−m b (ϕ), h).Proof.We have g ∈ ΣC n p,0,b (h 1 ; ϕ, h) equivalent to T n b (g) ∈ ΣC p,0 (h 1 ; T n b (ϕ), h), from Proposition 3. Since T n b (g) = T m b (T n−m b g) and T n b (ϕ) = T m b (T n−m b (ϕ)), we obtain