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The development of certain aspects of geometric function theory after incorporating fractional calculus and q-calculus aspects is obvious and indisputable. The study presented in this paper follows this line of research. New results are obtained by applying means of differential subordination and superordination theories involving an operator previously defined as the Riemann-Liouville fractional integral of the q-hypergeometric function. Numerous theorems are stated and proved involving the fractional q-operator and differential subordinations for which the best dominants are found. Associated corollaries are given as applications of those results using particular functions as best dominants. Dual results regarding the fractional q-operator and differential superordinations are also considered and theorems are proved where the best subordinants are given. Using certain functions known for their remarkable geometric properties applied in the results as best subordinant, interesting corollaries emerge. As a conclusion of the investigations done by applying the means of the two dual theories considering the fractional q-operator, several sandwich-type theorems combine the subordination and superordiantion established results.

In this paper, we discuss and introduce a new study using an integral operator w k , μ m in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In addition, certain sandwich theorems were found. The differential subordination theory's features and outcomes are symmetric to those derived using the differential subordination theory. [ABSTRACT FROM AUTHOR]

The development of certain aspects of geometric function theory after incorporating fractional calculus and q-calculus aspects is obvious and indisputable. The study presented in this paper follows this line of research. New results are obtained by applying means of differential subordination and superordination theories involving an operator previously defined as the Riemann- Liouville fractional integral of the q-hypergeometric function. Numerous theorems are stated and proved involving the fractional q-operator and differential subordinations for which the best dominants are found. Associated corollaries are given as applications of those results using particular functions as best dominants. Dual results regarding the fractional q-operator and differential superordinations are also considered and theorems are proved where the best subordinants are given. Using certain functions known for their remarkable geometric properties applied in the results as best subordinant, interesting corollaries emerge. As a conclusion of the investigations done by applying the means of the two dual theories considering the fractional q-operator, several sandwich-type theorems combine the subordination and superordiantion established results. [ABSTRACT FROM AUTHOR]

The results presented in this paper highlight the property of the Gaussian hypergeometric function to be a Carathéodory function and refer to certain differential inequalities interpreted in form of inclusion relations for subsets of the complex plane using the means of the theory of differential superordination and the method of subordination chains also known as Löwner chains.

In this paper, we discuss and introduce a new study using an integral operator wk,μm in geometric function theory, especially sandwich theorems. We obtained some conclusions for differential subordination and superordination for a new formula generalized integral operator. In addition, certain sandwich theorems were found. The differential subordination theory’s features and outcomes are symmetric to those derived using the differential subordination theory.

In the present article, we determine some subordination and superordination results involving Wanas operator for certain normalized analytic functions defined in the unit disk U. These results are applied to establish sandwich results. Our results extend corresponding previously known results.

In this present paper, we obtain some differential subordination and superordination results, by using generalized operators for certain subclass of analytic functions in the open unit disk. Also, we derive some sandwich results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to derive subordination and superordination results involving differential operator for multivalent analytic functions in the open unit disk. These results are applied to obtain sandwich results. Our results extend corresponding previously known results. [ABSTRACT FROM AUTHOR]

In this current study, we aim to give some results for third-order differential subordination and superordination for analytic functions in U={z∈ℂ:|z|<1} involving the generalized operator Iα,βjf. The results are derived by investigating relevant classes of admissible functions. Some new results on differential subordination and superordination with some sandwich theorems are obtained. Moreover, several particular cases are also noted. The properties and results of the differential subordination are symmetry to the properties of the differential superordination to form the sandwich theorems.

In the present paper, we obtain some subordination and superordination Results involving the integral operator Tα for certain normalized analytic functions in the open unit disk. These results are applied to obtain sandwich results. [ABSTRACT FROM AUTHOR]

Let Ω and Δ be any sets in ℂ, and let φ(r,s,t; z) : ℂ³ x U → ℂ Let p be a complex-valued harmonic function in the unit disc U of the form p(z) = p1 (z) + p2 (z), where p1 and p2 are analytic in U. In [5] the authors have determined properties of the function p such that p satisfies the differential subordination φ(p(z),Dp(z),D²p(z); z) C Ω ⇒‾ p(U) ⊂ Δ. In this article, we consider the dual problem of determining properties of the function p, such that p satisfies the second-order differential superordination Ω ⊂ φ(p(z), Dp(z),D²p(z); z) ⇒ Δ ⊂ p(U). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new di¤erential operator of analytic functions involving binomial series. Furthermore, we derive some subordination and superordination results for this operator. Some applications and examples are also obtained.

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied.

In the present investigation, by making use of strong differential subordinations and superordinations, we introduce and study two new classes of holomorphic functions containing generalized differential oper- ator. Also we determine important properties for functions belongs to these classes. [ABSTRACT FROM AUTHOR]

The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in 2015. In a recent paper published in 2019, the authors have considered the dual problem of the differential subordination for the harmonic complex-valued functions and have defined the differential superordination for harmonic complex-valued functions. Finding the best subordinant of a differential superordination is among the main purposes in this research subject. In this article, conditions for a harmonic complex-valued function p to be the best subordinant of a differential superordination for harmonic complex-valued functions are given. Examples are also provided to show how the theoretical findings can be used and also to prove the connection with the results obtained in 2015.

We obtain new outcomes of analytic functions linked with operator Hα,βη,k(f) defined by Mittag–Leffler function. Moreover, new theorems of differential sandwich-type are obtained.

In this present paper, we obtain some applications of first order differential subordination and superordination results involving Hadamard product for multivalent analytic functions with generalized hypergeometric function in the open unit disk. These results are applied to obtain sandwich results. Many of the well-known and new results are shown to follow as special cases of our results. [ABSTRACT FROM AUTHOR]

In this paper we present a few applications of differential subordinations. For functions p analytic in the open unit disc U, with p(0) = 1, consisting of p(z) is subordinate to q(z) = (1+ Az) /(1− Az) .

In this current study, we aim to give some results for third-order differential subordination and superordination for analytic functions in U = { z ∈ ℂ : | z | < 1 }   involving the generalized operator I α , β j f . The results are derived by investigating relevant classes of admissible functions. Some new results on differential subordination and superordination with some sandwich theorems are obtained. Moreover, several particular cases are also noted. The properties and results of the differential subordination are symmetry to the properties of the differential superordination to form the sandwich theorems. [ABSTRACT FROM AUTHOR]

S. S. Miller and P. T. Mocanu introduced the notion of differential superordination as a dual concept of differential subordination. The notion of strong differential subordination was introduced by J. A. Antonino and S. Romaguera. By using the Sălăgean differential operator we introduce a class of holomorphic functions denoted by Smn (α), and obtain some strong subordinations results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to derive some subordination and superordination results for certain normalized analytic functions in the open unit disk, acted upon by Carlson-Shaffer operator. Relevant connections of the results, which are presented in the paper, with various known results are also considered. [ABSTRACT FROM AUTHOR]

In the present investigation, we introduce new classes of p-valent meromorphic functions defined by Liu-Srivastava linear operator and the multiplier transform and study their properties by using certain first order differential subordination and superordination. [ABSTRACT FROM AUTHOR]

Let q 1, q 2 be univalent in Δ:= { z: | z| < 1} and p be certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization for various known sandwich theorems: implies q 1( z) ≺ p( z) ≺ q 2( z), where k ∈ ℤ and β ≠ 0, α′ j ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators. [ABSTRACT FROM AUTHOR]

The purpose of this present article is to derive some subordination and superordination results for certain normalized analytic functions in the open unit disk, acted upon by a class of multiplier transformations. [ABSTRACT FROM AUTHOR]

The purpose of this present article is to derive some subordination and superordination results involving Cho-Kim transformation for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in the article, with various known results are pointed out. [ABSTRACT FROM AUTHOR]

By making use of the familiar Carlson-Shaffer operator, the authors derive derive some subordination and superordination results for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in this paper, with various other known results are also pointed out. [ABSTRACT FROM AUTHOR]

In the present paper, we give, some applications of first order differential superordinations to obtain sufficient conditions for normalized analytic functions defined by certain linear operators to be superordinated to a given univalent function. [ABSTRACT FROM AUTHOR]

Both the theory of differential subordination and its dual, the theory of differential superordination, introduced by Professors Miller and Mocanu are based on reinterpreting certain inequalities for real-valued functions for the case of complex-valued functions. Studying subordination and superordination properties using different types of operators is a technique that is still widely used, some studies resulting in sandwich-type theorems as is the case in the present paper. The fractional integral of confluent hypergeometric function is introduced in the paper and certain subordination and superordination results are stated in theorems and corollaries, the study being completed by the statement of a sandwich-type theorem connecting the results obtained by using the two theories. [ABSTRACT FROM AUTHOR]

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach's conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied. [ABSTRACT FROM AUTHOR]

The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in 2015. In a recent paper published in 2019, the authors have considered the dual problem of the differential subordination for the harmonic complex-valued functions and have defined the differential superordination for harmonic complex-valued functions. Finding the best subordinant of a differential superordination is among the main purposes in this research subject. In this article, conditions for a harmonic complex-valued function p to be the best subordinant of a differential superordination for harmonic complex-valued functions are given. Examples are also provided to show how the theoretical findings can be used and also to prove the connection with the results obtained in 2015. [ABSTRACT FROM AUTHOR]

We obtain new outcomes of analytic functions linked with operator H α , β η , k (f) defined by Mittag–Leffler function. Moreover, new theorems of differential sandwich-type are obtained. [ABSTRACT FROM AUTHOR]