Abstract
This investigation deals with a new symmetric formula for a class of meromorphic analytic functions in the puncher open unit disk. Accordingly, the symmetric formula is employed to define a convolution linear operator associated with a special function type the incomplete hypergeometric function. By making utilize the recommended operators, a new sub-classes of meromorphic functions is presented discussing some subordination results. As exceptional examples of our results, certain more acknowledged outcomes are also provided.
Keywords: Meromorphic functions, Subordination, Analytic function, Differential operator, Open unit disk, Symmetric differential operator, Univalent function, Starlike function
1. Introduction
Suppose that is the class of meromorphic functions in , denoting by and taking the form
And assume that is the negative class of meromorphic functions in , such that
Remembering the attitude of subordination between analytic functions can be defined as follows: assume that the functions h and p are analytic in the unit disk, then the function h is subordinate to p if there occurs an analytic function in U achieving
This definition is represented by
Constructing usage of the definition of subordination in the space of holomorphic functions, Miller and Mocanu [1] and other researchers presented a lot of motivating concerning the differential subordination keeping the behaviors of well-known integral and differential operators (see [2], [3], [4], [5], [6], [7]). Additionally, Miller and Mocanu [1] considered the dual concept of differential subordinations, which is called the superordination (see also [6]). It should be commented that in current years, numerous authors found various remarkable outcomes concerning different integral operators connected with differential subordination and superordination. This concept is used to define classes of analytic functions, which are called Ma-Minda type classes [8]. In 2000, Polatoglu and Bolcal [9] furnished the boundaries of the radius of α-convexity for definite relations of analytic functions in U utilizing subordination concept and the authors in [10] achieved adequate conditions for functions being starlike of order α. Similarly, Baner [7] considered the geometric relations in the superior half plane and showing that they have parametric illustration utilizing subordination idea. Recent studies on the class of meromorphic functions are indicated in the efforts [11], [12], [13].
We have the following classes:
Definition 1.1
Let ϒ be analytic function in a range including and . Let be a specified analytic function in U. The function is known ϒ-like with respect to Ψ if
where (the necessary conditions). As a special case, when , we have
Definition 1.2
Let , where
and
under the conditions . Then the Hadamard product is defined by
and
In this effort, we shall define a new difference-differential symmetric operator for functions in the class and similarly in . We study the operator via different classes of analytic functions, especially the starlike sub-classes. Some consequences results are given.
2. Symmetric meromorphic operator (SMO)
The formula for the Symmetric Meromorphic Operator (SMO) of one dimensional parameter , which takes the class described in the preceding part, is covered in this section. The key step in discussing the operator geometrically is to demonstrate that the new symmetric operator formula belongs to the same class. In order to ensure that the coefficients have non-negative values, the constraints on α are computed. SMO's semi-group attribute is examined. The primary class of analytic functions known as Y-like involving SMO is given last.
Now we define a new meromorphic difference-differential operator of a complex variable by utilizing the class of meromorphic functions as follows. Let then
(2.1) |
Note that the term
may take a complex value. Therefore, the operator (2.1), indicates complex coefficients. To make sure that it is in the class we have a set of conditions on .
We have the following remark:
Remark 2.1
- •
Clearly, when then respectively, under some restriction on .
- •
By the definition of , we confirm thatobligates to the Hadamard product.
- •
It is clear that if and only if one of the following cases is achieved
- ▹
;- ▹
;- ▹
- ▹
- ▹
Over the reals number, we have .- •
For ♭ > 0, we have the following restrictions:
- ▹
;- ▹
;- ▹
;- ▹
.
In the sequel, we assume that satisfies one of the above conditions to be sure that . That is
Recently, there is attention towards the symmetric differential operators. Ibrahim and Darus presented a symmetric differential and integral operators for the class of the normalized analytic functions in the open unit disk [14]. Later, the researchers formulated different types of generalized symmetric differential operators [14], [15], [16].
Proposition 2.2
(Semi-group property) The SMO satisfies the semi-group property, as follows:
where .
Proof
Let . Then for , we have
By indication, we have the desire assertion. □
Our proofs are based on the following outcomes.
Lemma 2.3
(see[2]) Letbe convex univalent in the unit diskand κ andwith
Ifis holomorphic in Π and
then
and ω is the best dominant.
Lemma 2.4
(see[1]) Letbe univalent in U and ϑ and ϱ be holomorphic in a domain Γ containing with when . Set
Suppose that is starlike univalent in Π, and
for . If
then
and is the best dominant.
In this effort, we deal with the generalized class of meromorphic functions
Definition 2.5
Let ϒ be analytic function in a domain containing and . Let be a specified holomorphic function in U. The function is called ϒ-like with respect to ω if
(2.2) where (the necessary conditions). As a special case of (2.2), when , we have
Our aim is to illustrate a set of conditions that makes H satisfies (2.2).
3. Subordination inequalities
In this section, we obtain several adequate assumptions for subordination of holomorphic functions in the classes and . Through various classes of analytic functions, particularly the starlike subclasses, we investigate the operator. Results of some ramifications are provided. We deal with the subordination inequality in Definition 2.5. We shall illustrate a set of conditions that implies the relation when ω is starlike then SMO is also starlike. The consequences are discovered for special values of the parameters. Sharp results and the best dominated function are also presented.
Theorem 3.1
Assume that is convex univalent function in U with and
Suppose that is holomorphic in U. If fulfills the subordination
Then
and is the best dominant.
Proof
Suppose that is given by
Hence, we have
At that point, by the supposition of this result, we attain that the declaration of the result indicates an application of Lemma 2.3. □
For , in Theorem 3.1, we state the following corollary.
Corollary 3.2
Let the following subordination fulfilled
then
Again, let in Theorem 3.1, we state the next corollary.
Corollary 3.3
Suppose that the inequality
is satisfied, then
Theorem 3.4
Consider the function is univalent in U such that , and
is starlike univalent in U and
If satisfies
then
and is the best dominant.
Proof
Suppose that is given by
By setting and , it can clearly indicated that is analytic in , is analytic in . Also, we obtain
and
It is clear that is starlike univalent in U and
□
Taking in Theorem 3.4, we have the following outcome.
Corollary 3.5
If the subordination
satisfies then
We proceed to discuss multi-convoluted formula of the suggested operator taking the structure (3.1)
(3.1) |
where
and
We have the following result
Theorem 3.6
The functionsand W have the structure(3.1)if they satisfy
The functionprovided by achieves equality
Proof
From the inequality (3.1), we have
where is a Schwarz function. Now by assuming we have
A computation yields
Considering that , we obtain the assertion of this theorem. □
4. Linear operator associated with special functions
In this section, we introduce hybrid linear operators involving the SMO and special function terms. We shall introduce two types of hybrid linear operators associating with hypergeometric function and Mittag-Leffler function respectively. To make the connection, we request to make a change in the usual formula to fit the expression of SMO. Therefore, we present a modify expression to be sure that the new formula belongs to the main class of the analytic functions. The next step is convoluted SMO with the meromorphic hypergeometric function and Mittag-Leffler function. Proceeding with these convoluted operators, we formulate them in new classes of analytic functions of Y-type using the subordination inequality. To study them geometrically, we illustrate a set of conditions to get the starlikeness. Examples with graphs are presented in the sequel. Sharp and the best dominant results are also considered.
4.1. Hyper-geometric function
Define a special function analog to hypergeometric functions
where and is the Pochhammer symbol. Corresponding to , we define a linear operator utilizing the Hadamard product as follows:
(4.1) |
Note that, when in (4.1), we get the Carlson-Sheffer operator of a meromorphic function [17]. Moreover, it is clear that when , and that .
We define the following class
(4.2) |
where (the necessary conditions). As a special case of (4.2), when , we have
Theorem 4.1
Suppose that the functionis convex univalent in U withand
Also, assume thatis analytic in U. Ifachieves the inequality
Then
andis the best dominant.
Proof
Suppose that is given by
Hence, we have
Then by the conditions of the theorem we get that the desired affirmation using Lemma 2.3. □
Theorem 4.2
Consume that the function is univalent in U with is starlike univalent in U and
If fulfills
then
and is the best dominant.
Proof
Consider is given by
By setting and , it can easily observed that is analytic in , is analytic in . Also we obtain
and
Obviously, is starlike with a positive real structured as follows:
□
Example 4.3
Let such that
Then,
and for , we have
and when , we get
Fig. 1 indicates the behavior of H under for different values of , while Fig. 2 presents its derivative.
Figure 1.
The 3D and cantor plot of the function H(z − q) and its behavior under the suggested operator , where (A1-A2) indicated the graph of H(z − q), (B1-B2) presented the graph of Δ1 H(z − q), (C1-C2) plotted the graph of Δ0.5 H(z − q) and (D1-D2) provided the plot of Δ0.25 H(z − q) respectively.
Figure 2.
The 3D and cantor plot of H′(z − q) and its behavior under the suggested operator . From the left: (A1-A2) indicated the plot of H′(z − q), and (B1-B2) presented the graph of respectively.
Theorem 4.4
The functions and have the structure (3.1) if they satisfy
The function provided by achieves equality
Proof
From the inequality (3.1), we have
where is a Schwarz function. Now by assuming we have
A computation yields
Considering that , we obtain the assertion of this theorem. □
4.2. Mittag-Leffler function MLF
The following series indicate the MLF
In the same manner of the above instruction, we obtain
Corresponding to , we define a linear operator utilizing the Hadamard product as follows:
Note that, when , we get the Carlson-Sheffer operator of a meromorphic function [17]. Moreover, it is clear that when .
We define the following class
where (the necessary conditions). As a special case, when , we have
The following outcomes are similar to the above subsection, therefore, we ignore the proofs.
Theorem 4.5
Suppose that the functionis convex univalent in U withand
Also, assume thatis analytic in U. Ifachieves the inequality
Then
andis the best dominant.
Theorem 4.6
Consume that the functionis univalent in U withis starlike univalent in U and
Iffulfills
then
andis the best dominant.
Theorem 4.7
The functionsandhave the structure(3.1)if they satisfy
The functionprovided by achieves equality
5. Conclusion
The above study indicated several facts in the theory of the meromorphic analytic functions. We formulated new symmetric operators. The first one is based on a special class of meromorphic functions and the second one is a linear operator given by a special function type incomplete hypergeometric function. We formulated the sufficient conditions, in view of the subordination results, to prove that the operators are in ϒ-like class.
Our new method can be employed to describe the meromorphic solution of classes of algebraic differential equations of complex variables geometrically. This class of differential equations are very important in optics and other mathematical physics applications. From above investigation, we studied a class of first order differential inequality. Therefore, the meromorphic solution is indicated as an upper solution corresponding to its differential equation. For future works, may a researcher formulate the operator in terms of second order differential inequality using the second order differential subordination.
Funding statement
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
CRediT authorship contribution statement
Ibtisam Aldawish: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.
Declaration of Competing Interest
The authors declare no conflict of interest.
Data availability
Data included in article/supp. material/referenced in article.
References
- 1.Miller S.S., Mocanu P.T. CRC Press; 2000. Differential Subordinations: Theory and Applications. [Google Scholar]
- 2.Shanmugam T.N., Ravichandran V., Sivasubramanian S. Differential sandwich theorems for some subclasses of analytic functions. J. Aust. Math. Anal. Appl. 2006;3(1):1–11. [Google Scholar]
- 3.Ibrahim R.W., Aldawish I. Difference formula defined by a new differential symmetric operator for a class of meromorphically multivalent functions. Adv. Differ. Equ. 2021;2021(1):1–16. [Google Scholar]
- 4.Ibrahim R.W., Darus M. Sufficient conditions for subordination of meromorphic functions. J. Math. Stat. 2009;5(3):141. [Google Scholar]
- 5.Ibrahim R.W., Darus M. On operators for meromorphic functions. Int. Electron. J. Pure Appl. Math. 2011;3(2):139–145. [Google Scholar]
- 6.Bulbocã T. A class of superordination-preserving integral operators. Indag. Math. 2002;13:301–311. [Google Scholar]
- 7.Baner R.O. Chordal Lowener families and univalent Cauchy transforms. 2003. arXiv:math/0306130v1 [math.PR] pp. 1–29.
- 8.Ma W.C., Minda D. Proceedings of the Conference on Complex Analysis. 1992. A unified treatment of some special classes of univalent functions; pp. 157–169. [Google Scholar]
- 9.Polatoglu Y., Bolcal M. Some radius problems for certain families of analytic functions. Turk. J. Math. 2000;24:401–412. [Google Scholar]
- 10.Murugusundaramoorthy G., Janani T. Meromorphic parabolic starlike functions associated with-hypergeometric series. Int. Sch. Res. Not. 2014:2014. [Google Scholar]
- 11.Aouf M.K., El-Emam Fatma Z. Fekete–Szego problems for certain classes of meromorphic functions involving-Al-Oboudi differential operator. J. Math. 2022;2022 [Google Scholar]
- 12.Ibrahim Rabha W., Baleanu Dumitru, Jahangiri Jay M. Conformable differential operators for meromorphically multivalent functions. Concr. Oper. 2021;8(1):150–157. [Google Scholar]
- 13.Al-Khafaji Aqeel Ketab. On subclass of meromorphic analytic functions defined by a differential operator. J. Phys. Conf. Ser. 2021;1818(1) IOP Publishing. [Google Scholar]
- 14.Ibrahim R.W., Darus M. New symmetric differential and integral operators defined in the complex domain. Symmetry. 2019;11(7):906. [Google Scholar]
- 15.Ibrahim R.W., Elobaid R.M., Obaiys S.J. Symmetric conformable fractional derivative of complex variables. Mathematics. 2020;8(3):363. [Google Scholar]
- 16.Ibrahim R.W., Elobaid R.M., Obaiys S.J. Geometric inequalities via a symmetric differential operator defined by quantum calculus in the open unit disk. J. Funct. Spaces. 2020;2020 [Google Scholar]
- 17.Carlson B.C., Shaffer D.B. Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal. 1984;15(4):737–745. [Google Scholar]
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