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In this paper, the research discussed involves fractional calculus applied to a $ q $-operator. Fractional integrals applied to the $ q $-analogue of the multiplier transformation gives a new operator, and the research is conducted applying the differential subordination and superordination theories. The best dominant and the best subordinant are obtained by the theorems and corollaries discussed. Combining the results from the both theories, sandwich-type results are presented as a conclusion of this research.

In this paper, the research discussed involves fractional calculus applied to a q -operator. Fractional integrals applied to the q -analogue of the multiplier transformation gives a new operator, and the research is conducted applying the differential subordination and superordination theories. The best dominant and the best subordinant are obtained by the theorems and corollaries discussed. Combining the results from the both theories, sandwich-type results are presented as a conclusion of this research. In this paper, the research discussed involves fractional calculus applied to a -operator. Fractional integrals applied to the -analogue of the multiplier transformation gives a new operator, and the research is conducted applying the differential subordination and superordination theories. The best dominant and the best subordinant are obtained by the theorems and corollaries discussed. Combining the results from the both theories, sandwich-type results are presented as a conclusion of this research. [ABSTRACT FROM AUTHOR]

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries. [ABSTRACT FROM AUTHOR]

The present research aims to present new results regarding the fundamental problem of providing sufficient conditions for finding the best subordinant of a third-order differential superordination. A theorem revealing such conditions is first proved in a general context. As another aspect of novelty, the best subordinant is determined using the results of the first theorem for a third-order differential superordination involving the Gaussian hypergeometric function. Next, by applying the results obtained in the first proved theorem, the focus is shifted to proving the conditions for knowing the best subordinant of a particular third-order differential superordination. Such conditions are determined involving the properties of the subordination chains. This study is completed by providing means for determining the best subordinant for a particular third-order differential superordination involving convex functions. In a corollary, the conditions obtained are adapted to the special case when the convex functions involved have a more simple form. [ABSTRACT FROM AUTHOR]

The study presented in this article involves q-calculus connected to fractional calculus applied in the univalent functions theory. Riemann-Liouville fractional integral of q-hypergeometric function is defined here, and investigations are conducted using the theories of differential subordination and superordination. Theorems and corollaries containing new subordination and superordination results are proved for which best dominants and best subordinants are given, respectively. As an application of the results obtained by the means of the two theories, the statement of a sandwich-type theorem concludes the study.

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.

The results obtained by the authors in the present article refer to quantum calculus applications regarding the theories of strong differential subordination and superordination. The q-analogue of the multiplier transformation is extended, in order to be applied on the specific classes of functions involved in strong differential subordination and superordination theories. Using this extended q-analogue of the multiplier transformation, a new class of analytic normalized functions is introduced and investigated. The convexity of the set of functions belonging to this class is proven and the symmetry properties derive from this characteristic of the class. Additionally, due to the convexity of the functions contained in this class, interesting strong differential subordination results are proven using the extended q-analogue of the multiplier transformation. Furthermore, strong differential superordination theory is applied to the extended q-analogue of the multiplier transformation for obtaining strong differential superordinations for which the best subordinants are provided. [ABSTRACT FROM AUTHOR]

The results obtained by the authors in the present paper refer to quantum calculus applications regarding the theories of differential subordination and superordination. These results are established by means of an operator defined as the q-analogue of the multiplier transformation. Interesting differential subordination and superordination results are derived by the authors involving the functions belonging to a new class of normalized analytic functions in the open unit disc U, which is defined and investigated here by using this q-operator. [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 present research aims to present new results regarding the fundamental problem of providing sufficient conditions for finding the best subordinant of a third-order differential superordination. A theorem revealing such conditions is first proved in a general context. As another aspect of novelty, the best subordinant is determined using the results of the first theorem for a third-order differential superordination involving the Gaussian hypergeometric function. Next, by applying the results obtained in the first proved theorem, the focus is shifted to proving the conditions for knowing the best subordinant of a particular third-order differential superordination. Such conditions are determined involving the properties of the subordination chains. This study is completed by providing means for determining the best subordinant for a particular third-order differential superordination involving convex functions. In a corollary, the conditions obtained are adapted to the special case when the convex functions involved have a more simple form.

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries.

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 the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results. [ABSTRACT FROM AUTHOR]

In the research presented in this paper, confluent hypergeometric function is embedded in the theory of strong differential superordinations. In order to proceed with the study, the form of the confluent hypergeometric function is adapted taking into consideration certain classes of analytic functions depending on an extra parameter previously introduced related to the theory of strong differential subordination and superordination. Operators previously defined using confluent hypergeometric function, namely Kummer–Bernardi and Kummer–Libera integral operators, are also adapted to those classes and strong differential superordinations are obtained for which they are the best subordinants. Similar results are obtained regarding the derivatives of the operators. The examples presented at the end of the study are proof of the applicability of the original results.

In this paper, we aim to present a survey on subordination and superordination theorems related to the class of analytic functions defined in a symmetric domain, which is the open unit disc. The results were deduced by making use of a new differential operator. We present two properties of this operator from which we constructed the final results. Moreover, based on the obtained outcomes, we give two sandwich-type theorems. Some interesting further consequences are also taken into consideration.

The notion of strong differential subordination was introduced in 1994 and the theory related to it was developed in 2009. The dual notion of strong differential superordination was also introduced in 2009. In a paper published in 2012, the notion of strong differential subordination was given a new approach by defining new classes of analytic functions on U×U¯ having as coefficients holomorphic functions in U¯. Using those new classes, extended Sălăgean and Ruscheweyh operators were introduced and a new extended operator was defined as Lαm:Anζ*→Anζ*,Lαmf(z,ζ)=(1−α)Rmf(z,ζ)+αSmf(z,ζ),z∈U,ζ∈U¯, where Rmf(z,ζ) is the extended Ruscheweyh derivative, Smf(z,ζ) is the extended Sălăgean operator and Anζ*={f∈H(U×U¯),f(z,ζ)=z+an+1ζzn+1+⋯,z∈U,ζ∈U¯}. This operator was previously studied using the new approach on strong differential subordinations. In the present paper, the operator is studied by applying means of strong differential superordination theory using the same new classes of analytic functions on U×U¯. Several strong differential superordinations concerning the operator Lαm are established and the best subordinant is given for each strong differential superordination.

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]

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 this paper we study certain strong differential superordinations, obtained by using a new integral operator introduced in [13]. [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.

In the present paper we establish several differential super ordinations regarding the operator DRmλ defined by using Ruscheweyh derivative Rm f (z) and the generalized Salagean operator Dmλ f (z), DRmλ : An → An, DRmλ f (z) = (Dmλ * Rm) f (z) ; z ϵ U; where m; n ϵ N, λ ⩾ 0 and f ∊ An; An = ff 2 H(U) : f(z) = {f ∊ H(U): f(z)=z+Σjj ; ∞ j=n+1: ajzj, z ϵ U}. A number of interesting consequences of some of these super ordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided. [ABSTRACT FROM AUTHOR]

In the present paper we establish several strong differential superordinations regarding the extended multiplier transformation I (m, λ, l): Anζ* → Anζ*, I (m, λ, l) f (z, ζ) = z+∑j=n+1∞ (1+λ(j-1)+1/l+1)m aj(ζ)zj, z ∈ U, ζ ∈ ..., and Anζ* = (f ∈ ...(U x ...), f(z, ζ) = z + an+1(ζ) zn+1 + ..., z ∈ U, ζ ∈ ...} is the class of normalized analytic functions. [ABSTRACT FROM AUTHOR]

In the present paper we establish several differential superordinations regarding the operator RDλ,αn defined by using Ruscheweyh derivative Rn f (z) andthe generalizedSălăgean operator Dλn f (z), RDλ,αn : A → A, RD RDλ,αn f (z) = (1 - α)Rn f (z) + αDλn f (z),z ∈ - U, where n ∈ ..., λ, α ≥ 0 and f ∈ A, A = {f ∈ H(U) : f (z) = z + ...,z ∈ U}. A number of interesting consequences of some of these superordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided. [ABSTRACT FROM AUTHOR]

In the present paper we define a new operator using the generalized Sălăgean and Ruscheweyh operators. Denote by DRmλ the Hadamard product of the generalized Sălăgean operator Dmλ and the Ruscheweyh operator Rm, given by DRmλ : An ℯ An, DRmλ f(z) = (Dmλ * Rm) f(z) and An = {f ∈ H(U); f(z) = z + an + 1zn+1 + ..., z ∈ U} is the class of normalized analytic functions. We study some differential superordinations regarding the operator DRmλ. [ABSTRACT FROM AUTHOR]

In this paper we study certain strong differential superordinations and give a sandwich theorem, obtained by using a new integral operator introduced in [21]. [ABSTRACT FROM AUTHOR]

In this paper we study certain strong differential subordinations and strong differential superordinations; obtained by using a new integral operator introduced in [21]. We also give some results as a sandwich theorem. [ABSTRACT FROM AUTHOR]

In this paper, we aim to present a survey on subordination and superordination theorems related to the class of analytic functions defined in a symmetric domain, which is the open unit disc. The results were deduced by making use of a new differential operator. We present two properties of this operator from which we constructed the final results. Moreover, based on the obtained outcomes, we give two sandwich-type theorems. Some interesting further consequences are also taken into consideration. [ABSTRACT FROM AUTHOR]

In the present paper we establish several strong differential superordinations regardind the new operator DRλm defined by convolution product of the extended Sălăgean operator and Ruscheweyh derivative, DRλm : A*nζ → A*nζ, DRλm f (z, ζ) = (DRλm Rm) f (z, ζ) , z ∈ U, ζ ∈ U, where Rmf(z, ζ) denote the extended Ruscheweyh derivative, Dm λ f(z, ζ) is the extended generalized Sălăgean operator and A nζ = {f ζ H(U × Ū), f(z, ζ) = z + an+1 (ζ) zn+1 + … , z ζ U, ζ ∈ U} is the class of normalized analytic functions. [ABSTRACT FROM AUTHOR]

Abstract: Let and belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions to satisfy the double subordination chain . The differential sandwich-type result obtained is applied to normalized univalent functions and to -like functions. [Copyright &y& Elsevier]

In the present paper we establish several strong differential superordinations regardind the new operator DR Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. defined by convolution product of the extended generalized Sălăagean operator and Ruscheweyh derivative, DR Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. : 효 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. → 효 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. DR Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (z, ζ ) = (D Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.* Rm f (z; ζ) ; where Rm f(z, ζ) denote the extended Ruscheweyh derivative, D Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. f(z, ζ) is the extended generalized Sălăgean operator and 효 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.ζ = {f ∈ 횮(U x Ū), f(z, ζ) = z + an+1 (ζ) z n+1 +…, z ∈ U, ζ, ∈ Ū}, with 효 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.ζ = 효 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., is the class of normalized analytic functions. [ABSTRACT FROM AUTHOR]

In the present paper we establish several strong differential superordinations regardind the new operator SRm defined by convolution product of the extended Sălăgean operator and Ruscheweyh derivative, SRm : 효Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.ζ → 효 Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. ζ, SRmf (z, ζ) = Sm * Rm) f (z, ζ), z ∈ U, ζ ∈ Ū, where Rm f(z, ζ) denote the extended Ruscheweyh derivative, Sm f(z; ζ) is the extended Sălăagean operator and 효Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.ζ = {f ∈ 횮(U x Ū); f(z,ζ) = z + an+1 (ζ) z n+1 +…, z ∈ U, ζ ∈ Ū is the class of normalized analytic functions. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of the present paper is to obtain some sandwich-type theorems, which are combined by subordination and superordination preserving properties associated with a certain family of multiplier transformations for meromorphic multivalent functions in the punctured open unit disk. [Copyright &y& Elsevier]

Abstract: In the present paper we establish several differential superordinations regarding the new operator defined by using the generalized Sălăgean operator and Ruscheweyh derivative , , where , and , A number of interesting consequences of some of these superordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided. [Copyright &y& Elsevier]

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]

In the present paper we define a new operator, by means of convolution product between Ruscheweyh operator and the multiplier transformation I (m, λ, l). For functions f belonging to the class An we define the differential operator IRmλ,l : An → An, IRmλ,lf (z) := (I (m, λ, l) * Rm) f (z) , where An = {f ϵ H(U) : f(z) = z+an+1zn+1+…, z ϵ U} is the class of normalized analytic functions. We study some differential superordinations regarding the operator IRm λ,l. [ABSTRACT FROM AUTHOR]

Using the methods of differential subordination and superordination, sufficient conditions are determined on the multiplier transformation for meromorphic functions in the punctured unit disk to obtain respectively the best dominant and best subordinant. New differential sandwich-type results are also obtained for this multiplier transformation. [ABSTRACT FROM AUTHOR]

The notion of strong differential subordination was introduced in 1994 and the theory related to it was developed in 2009. The dual notion of strong differential superordination was also introduced in 2009. In a paper published in 2012, the notion of strong differential subordination was given a new approach by defining new classes of analytic functions on U × U ¯ having as coefficients holomorphic functions in U ¯ . Using those new classes, extended Sălăgean and Ruscheweyh operators were introduced and a new extended operator was defined as L α m : A n ζ * → A n ζ * , L α m f (z , ζ) = (1 − α) R m f (z , ζ) + α S m f (z , ζ) , z ∈ U , ζ ∈ U ¯ , where R m f (z , ζ) is the extended Ruscheweyh derivative, S m f (z , ζ) is the extended Sălăgean operator and A n ζ * = { f ∈ H (U × U ¯) , f (z , ζ) = z + a n + 1 ζ z n + 1 + ⋯ , z ∈ U , ζ ∈ U ¯ }. This operator was previously studied using the new approach on strong differential subordinations. In the present paper, the operator is studied by applying means of strong differential superordination theory using the same new classes of analytic functions on U × U ¯. Several strong differential superordinations concerning the operator L α m are established and the best subordinant is given for each strong differential superordination. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of the present paper is to investigate several subordination- and superordination-preserving properties of a certain class of integral operators, which are defined on the space of meromorphic functions in the punctured open unit disk. The sandwich-type theorem for these integral operators is also presented. Moreover, we consider an application of the subordination and superordination theorem to the Gauss hypergeometric function. [Copyright &y& Elsevier]

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]