Subordination Results for the Second-Order Differential Polynomials of Meromorphic Functions

: The outcome of the research presented in this paper is the deﬁnition and investigation of two new subclasses of meromorphic functions. The new subclasses are introduced using a differential operator deﬁned considering second-order differential polynomials of meromorphic functions in U \{ 0 } = { z ∈ C : 0 < | z | < 1 } . The investigation of the two new subclasses leads to establishing inclusion relations and the proof of convexity and convolution properties regarding each of the two subclasses. Further, involving the concept of subordination, the Fekete–Szegö problem is also discussed for the aforementioned subclasses. Symmetry properties derive from the use of the convolution and from the convexity proved for the new subclasses of functions.


Introduction
The results presented in this paper are obtained considering the general context of geometric function theory and involve the class of meromorphic functions, a differential operator and certain well-known and intensely used tools for investigation, namely the concepts of subordination and convolution.
The basic classes involved in this study are introduced in the unit disc of the complex plane U = {z ∈ C : |z| < 1}.
Let A denote the class of analytic functions in U of the form which satisfies the relations f (0) = 0, f (0) = 1. The subclass of A containing the univalent functions in U is denoted by S. The subclasses of A referred to as the class of starlike functions and the class of convex functions are denoted by S * and K, respectively, and are defined as S * = f ∈ A : Re z f (z) f (z) > 0 and K = f ∈ A : Re z f (z) f (z) + 1 > 0 . Another special class of analytic and univalent functions is the class of meromorphic functions denoted by Σ and containing functions of the form: a n z n = 1 z + a 1 z + a 2 z 2 + a 3 z 3 + . . . , (2) analytic and univalent in U * = U\{0} = {z ∈ C : 0 < |z| < 1}.
Following the same procedure as seen in [17] and used by many other authors, the following operator is defined: Definition 2. Considering f ∈ Σ and f (z) given by (6), a new differential operator is defined as: Then, from (8) and (9), we see that For the investigation presented in this paper, the following two subclasses of meromorphic functions are introduced using the operator given by (10): In conjunction with (3) and (10), Definition 4. In conjunction with (4) and (10), In order to obtain the new results contained in Section 3, the following lemmas are necessary: Lemma 1 ([18]). Let p be analytic in U with p(0) = 1, and suppose that
Having in mind the very recent papers where the well-known Fekete-Szegö inequalities are evaluated for different subclasses of meromorphic functions [19][20][21], in Section 4 of the paper, the Fekete-Szegö problem is investigated concerning the classes defined by (11) and (12). In order to conduct the study, the notion of subordination is applied, and the classes are redefined in terms of subordination.
Let f and g be two analytic functions in U. We say that f is subordinate to g, written f (z) ≺ g(z), if there exists a Schwarz function w, analytic in U with w(0) = 0 and |w(z)| < 1 such that f (z) = g(w(z)); see [22]. If g is univalent in U, then f (z) ≺ g(z) is equivalent to f (0) = g(0) and f (U) ⊂ g(U).

Definition 5.
Consider Φ(z) an analytic function with ReΦ(z) > 0, z ∈ U, satisfying Φ(0) = 1, Φ (0) > 1. Such a function maps U onto a region that is symmetric with respect to the real axis and is starlike with respect to 1. Denote by S * ,k (Φ, λ) the class of functions f ∈ S that satisfy the subordination given by: Definition 6. Consider Φ(z) an analytic function with ReΦ(z) > 0, z ∈ U, satisfying Φ(0) = 1, Φ (0) > 1. Such a function maps U onto a region which is symmetric with respect to the real axis and is starlike with respect to 1. Denote by C k (Φ, λ) the class of functions f ∈ S that satisfy the subordination given by: We also note that For obtaining the Fekete-Szegö inequalities related to the classes given in Definitions 5 and 6, the following lemmas are applied: . is a function with a positive real part in U and σ ∈ R, then c 2 − σc 2 1 ≤ 2 max{1; |2σ − 1|}.

Lemma 4 ([23]
). If p(z) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + . . . is a function with a positive real part in U and σ ∈ R, then Now, after all the preliminary notations, definitions and lemmas are listed, the next two sections contain the new results, which the authors want to bring to researchers' attention. In Section 3, containment relations are established for the classes S * ,k (λ) and C k (λ) introduced in Definitions 3 and 4. It is proved that the functions belonging to those classes are convex, and also, convolution properties are obtained using the functions f , g ∈ S * ,k (λ) and f , g ∈ C k (λ), respectively. In Section 4, the Fekete-Szegö problem is considered for the classes S * ,k (Φ) and C k (Φ) seen in Definitions 5 and 6.

Inclusion and Convolution Theorems
The results presented in this section refer to inclusion relations established for the classes S * ,k (λ) and C k (λ) introduced in Definitions 3 and 4, respectively. Convexity properties are stated for classes S * ,k (λ) and C k (λ), and convolution properties involving functions from the two classes are also proved easily by using the iterative-type operator seen in Definition 2.
The first two theorems from this section include containment relations obtained for classes S * ,k (λ) and C k (λ) given by (12) and (13).
Proof. Let z 2 f (z) ∈ S * ,k+1 (λ) and suppose that Set The analytic function p(z) satisfies conditions p(0) = 1 and p(z) = 0 for all z ∈ U. Differentiating logarithmically (18), and after manipulations, we obtain Now, by applying Lemma 1, we obtain that Proof. Applying (15) and Theorem 1, the following can be written: which evidently proves Theorem 2.
The next two theorems prove the property of the sets of functions S * ,k (λ) and C k (λ) to be convex.
Proof. Consider the function z 2 f (z) ∈ S * ,k+1 (λ) and suppose that belongs to the class S * ,k (λ). The proof requires to show that the function h(z) = u 1 f 1 (z) + u 2 f 2 (z), with u 1 and u 2 nonnegative and u 1 + u 2 = 1, belongs to the class S * ,k (λ).
Proof. From Theorem 3 and (15), it follows easily that Theorem 4 is true.
In the next four theorems, results related to convolution properties for the classes S * ,k (λ) and C k (λ) are derived.

Proof.
For z 2 f (z) ∈ S * ,k (λ), we write the following: where w is analytic in U with |w(z)| < 1 and w(z) = 0. From this, we obtain which upon integration yields Assertion (27) can easily be obtained from (26). and where w is analytic in U with |w(z)| < 1 and w(z) = 0.
Proof. Suppose that z 2 f (z) ∈ C k (λ); then (12) can be written as follows: where w is analytic in U with |w(z)| < 1 and w(z) = 0. From this, we obtain .
By integrating the above relation, we obtain: It follows that The equality given by relation (28) of Theorem 6 is easily obtained from (18) and (31).
Proof. Knowing that g (z) is convex univalent in U, from (11), we have By applying convolution properties, we deduce: ) .
The proof is concluded by applying Lemma 2.

Fekete-Szegö Problem
Fekete-Szegö inequalities are obtained in this section considering functions from classes S * ,k (Φ) and C k (Φ) given by Definitions 5 and 6, respectively. In order to obtain those results, similar methods to those seen in [24] are implemented, and the proof of the results is facilitated by the use of the iterative-type operator given by Definition 2. Theorem 9. Let Φ(z) = 1 + B 1 z + B 2 z 2 + B 3 z 3 + . . . , with (B 1 = 0) and f given in (6) belonging to S * ,k (Φ, λ); then, for σ ∈ C, The result is sharp.

Proof.
Consider the function f (z) ∈ S * ,k (Φ, λ). In this case, an analytic Schwarz function w exists with w(0) = 0 and |w(z)| < 1 in U such that We define the function p(z) by since w(z) is a Schwartz function. Therefore, By substituting (34) in (36), we obtain From Equation (37), we obtain Therefore, where The proof is concluded by applying Lemma 3. The sharpness of the results is obtained for the functions All the assertions of Theorem 9 are now proved.
) and let f given in (6) belong to C k (Φ, λ). Then, for σ ∈ C, The result is sharp.

Proof.
Consider the function f (z) ∈ C k (Φ, λ). In this case, an analytic Schwarz function w exists with w(0) = 0 and |w(z)| < 1 in U such that Along Equations (35), (36) and (40), we have From Equation (41), we obtain Therefore, where The proof is concluded by applying Lemma 3. Sharpness of the results is obtained for the functions This completes the proof of Theorem 10.
Taking λ = 0 in Theorems 9 and 10, we obtain the following results for functions belonging to the classes S * ,k (Φ) and C k (Φ).

Sharpness of the results is obtained for the functions
. 0). If f given in (6) belongs to the class C k (Φ), then, for σ, a real number, Sharpness of the results is obtained for the functions − 1 + z(D k z 2 f (z)) (D k z 2 f (z)) = Φ(z 2 ) and − 1 + z(D k z 2 f (z)) (D k z 2 f (z)) = Φ(z).
If f given by (6) belonging to S * ,k (Φ), then, for σ, a real number, where If f given by (6) belongs to C k (Φ), then, for σ, a real number,

Conclusions
After a few aspects regarding the lines of research involving meromorphic functions are highlighted, the investigation presented in this paper starts with the introduction of a new differential operator given in Definition 2. This operator is obtained using the second-order differential polynomials of meromorphic functions seen in Definition 1. A new subclass of meromorphic starlike functions S * ,k (λ) and a new subclass of meromorphic convex functions C k (λ) are introduced in Definitions 3 and 4, respectively. The new subclasses are investigated in Section 3 for inclusion relations, convexity and convolution properties. Using the concept of subordination, classes S * ,k (λ) and C k (λ) are redefined in Definitions 5 and 6, and the notations used for them become S * ,k (Φ) and C k (Φ) with reference to the subordinating function Φ instead of the parameter λ. The study is completed by establishing in Section 4 Fekete-Szegö inequalities regarding the coefficients of the functions from classes S * ,k (Φ) and C k (Φ).
As future directions of research where the results presented in this paper could be used, the connection between convexity properties and symmetry could be further explored. The convolution properties proved here suggest future studies where functions from the new subclasses S * ,k (λ) and C k (λ) could be combined with other functions with remarkable geometric and symmetry properties. The means of the theory of differential subordination could be used for further investigations on the two subclasses S * ,k (Φ) and C k (Φ), which could provide interesting subordination results with a nice geometric interpretation. The results obtained for multivalent meromorphic functions connected with the Liu-Srivastava operator using the theory of strong differential subordination [25] suggest that this theory could also be applied for the functions of classes S * ,k (λ) and C k (λ) involving the new operator defined here. Additionally, considering the strong differential results obtained for Sȃlȃgean and Ruscheweyh differential operators [26], the idea of applying the means of strong differential subordination to the differential operator given in Definition 2 seems interesting.