Abstract

Newly, numerous investigations are considered utilizing the idea of parametric operators (integral and differential). The objective of this effort is to formulate a new 2D-parameter differential operator (PDO) of a class of multivalent functions in the open unit disk. Consequently, we formulate the suggested operator in some interesting classes of analytic functions to study its geometric properties. The recognized class contains some recent works.

1. Introduction

In analysis, a parametric differential operator (PDO) is a differential operator of a dependent variable with respect to another dependent variable that is engaged when both variables formulate on an independent third variable, typically supposed as “time.” We shall use this idea to consider the PDO of a complex variable to discuss its properties in the opinion of the geometric function theory (GFT). The field of differential operators is investigated in GFT early by the well-known Salagean differential operator and the Ruscheweyh derivative. Later, these operators are generalized by different types of parameters using a 1D-parameter fractional differential operator [1] and 2D-parameter fractional differential operator [2]. Recently, using the class of normalized functions

Ibrahim and Jay [3] presented PDO of the following form: for

The functions are analytic in satisfying .

More studies are given by Ibrahim and Baleanu [4, 5] using (2) to present a hybrid diff-integral operator and a quantum hybrid operator, respectively.

In this effort, we generalize (2) by considering another class of analytic functions denoting by and constructing by which are analytic in Recently, different investigations are presented studying the geometric behavior of this class (see [6–9]).

The Hadamard product [10, 11] for two functions in is given by the series

Definition 1. For a functionPDO is defined as follows:where and are defined in (3) and (4), respectively.

Remark 2. (i)It is clear that , and it is a generalization of (2) ()(ii)The integral operator that corresponds to iswhere

Moreover, we have the following property:

Proposition 3 (semigroup property). Consider the PDO; then for and

Proof. Let ; the definition of implies Hence, for all , we have the desired assertion.

Our study is about the following class:

Definition 4. A functionis called in the classif it satisfies the inequalitywhere the symbol presents the subordination symbol [12] and is convex univalent in .

For example which is univalent convex in , and it is the extreme function in the set

Define a functional as follows: where

Shortly, by Definition 4 we say

Our aim is to study the operator formula . We recall the following results:

Lemma 5 (see [12]). Let two analytic functionsandbe convex univalent defined insuch thatMoreover, for a constantthe subordinationimplies

Lemma 6 (see [12]). Define the general class of holomorphic functionswhere and is a positive integer. If , then Moreover, if and then there are fixed numbers and with the inequality

Lemma 7 (see [13]). Let, whereis convex univalent inand forthen

Lemma 8 (see [14]). Let, whereis convex univalent insuch thatis univalent; then

Lemma 9 (see [15]). Let, andis convex univalent insuch thatand; then

2. The Results

In this section, we illustrate our main results concerning the class for some special .

2.1. General Properties

Theorem 10. Suppose that. Ifthen the coefficient bounds ofsatisfy the inequalitywhere is a probability measure. Also, if then , that is

Proof. By the assumption, we have Thus, the Carathéodory positivist technique yields where is a probability measure. In addition, if then according to Theorem 1.6 in [10] and for fixed we have Hence,
The next results show the sufficient and necessary conditions for the sandwich behavior of the functional

Theorem 11. Let the following assumptions holdwhere and convex in Moreover, let be univalent in such that where represents the set of all injection analytic functions with and Then and is the best subdominant, and is the best dominant.

Proof. Since then we obtain the next double inequality Thus, Lemmas 7 and 8 imply the desired assertion.

Theorem 12. Letbe a univalent convex function insuch thatandThen

Proof. By the definition of and clearly we have Hence, a direct application of Lemma 9, we obtain the result.

2.2. Inclusion Properties

In this part, we deal with the inclusion properties.

Theorem 13. Forandthen

Proof. Let Define the analytic function in , as follows: satisfying A computation gives Consequently, we get the inequality Applying Lemma 5 with gives Since and is convex univalent in , we arrive at the inequality Hence, by Definition 4, we conclude that

Theorem 14. LetThen where

Proof. A calculation implies that According to Lemma 6 joining the value we get

Corollary 15. Letbe assumed as in Theorem14. If the subordinationwhere holds, then

Proof. Taking, in Theorem 14 implies that Consequently, we have

Theorem 16. LetandIfthen

Proof. A convolution product indicates that where In view of real inequality (55), we get that has the Herglotz integral formula [11]. where conforms the probability measure on the unit circle and But, is convex in then we have Thus,

2.3. Fekete-Szegö Inequality

In this section, we obtain the Fekete-Szegö relation coefficient estimates for the class Let be the class of functions of the form in the open unit disk satisfying To prove our results, we need the following lemma.

Lemma 17 (see [16]). If, then for any complex numberThe result is sharp for the functions given by or

Theorem 18. Let the functionbe formulated by ((5)). Then,andwhere

Proof. Since we have In addition, there is a Schwarz function in such that Now by (18), we have where is given by (19). Equating the coefficients of and , we get From (67) and (69), we get For any , we get where By applying Lemma 17, we get The result is sharp for the function or

Remark 19. By fixingin Theorem18, we getwhere From Definition 4, a function is said to be in the class if it satisfies the inequality (13); then we have where is as given in (17).

Now, we obtain coefficient estimates for

Theorem 20. Let the functionbe defined by ((5)). Then,ifwhere is given by (8).

Proof. Suppose satisfies (80). Then, for

3. An Application

In this section, we consider the suggested class for all

Theorem 21. Consider the class of analytic functionsThen, the solution of the differential equation corresponds to this class iswhere represents the hypergeometric function.

Proof. Suppose that Then, it satisfies the differential equation where and This leads to the solution To find the upper solution, we let Thus, we have the differential equation Rewrite the above equation as follows: Multiplying the above equation by the functional we obtain Hence, it follows the solution (26).

Example 1. For(i) the solution is(ii); the solution becomes(iii); then the solution is given by the formula

4. Conclusion

Commencing overhead, we formulated a new parametric differential operator for a certain class of multivalently analytic functions. We investigated some geometric conducts of the operator connecting with the Janowski function, which is convex univalent in the open unit disk. As an application, we presented the formula of the suggested class involving the operator. For future works, one can generalize the suggested fractional operator using various classes of analytic functions such as meromorphic and harmonic functions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally and significantly to writing this article. All authors read and agreed to the published version of the manuscript.