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In this paper the authors combine the quantum calculus applications regarding the theories of differential subordination and superordination with fuzzy theory. These results are established by means of an operator defined as the q -analogue of the multiplier transformation. Interesting fuzzy 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 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]

In the present paper we establish several fuzzy differential subordinations regardind the operator I (m, λ l), given by I (m, λ l) : A → A, I (m, λ l) f (z) = z + Wε1;∞ j=ε (1+λ(j-1)+l /l+1)m ajzj and A = {f ε H(U), f(z) = z + a2z2 +…z … U} is the class of normalized analytic functions. A certain fuzzy class, denoted by SIδF (m, λ l), of analytic functions in the open unit disc is introduced by means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties and characteristics of the class SIδF (m, λ l) : Also, several fuzzy differential subordinations are established regarding the operator I (m; λ l). [ABSTRACT FROM AUTHOR]

In the present paper we establish several fuzzy differential subordinations regarding the operator L α m , given by L α m : A → A , L α m f ( z ) = ( 1 − α ) R m f ( z ) + α S m f ( z ) , where R m f ( z ) denote the Ruscheweyh derivative, S m f ( z ) is the Sălăgean operator and A n = { f ∈ H ( U ) , f ( z ) = z + a n + 1 z n + 1 + ⋯ , z ∈ U } is the class of normalized analytic functions with A 1 = A . A certain fuzzy class, denoted by S L F m ( δ , α ) , of analytic functions in the open unit disk is introduced by means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties and characteristics of the class S L F m ( δ , α ) . Also, several fuzzy differential subordinations are established regarding the operator L α m . [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 establish several fuzzy differential subordinations regarding the operator defined as Hadamard product of Salagean operator Sm and Ruscheweyh derivative Rm. denoted SRm. given by SRm : A → A, SRm f (z) = (Sm " Rm) f (z) and An = (f ∈ H(U), f(z) = z + an+1zn+1 + ..., z ∈ U) is the class of normalized analytic functions with A1 = A. A certain fuzzy class, denoted by SRFm (δ), of analytic functions in the open unit disc is introduced by means of this operator. By making use of the concept, of fuzzy differential subordination we will derive various properties and characteristics of the class SRFm (δ). Also, several fuzzy differential subordinations are established regarding the operator SRm. [ABSTRACT FROM AUTHOR]

In the present paper we establish several fuzzy differential subordinations regardind the operator RDmλ,α. given by RDmλ,α : An -→ An, RDmλ,α = (l-α)Rmf(z)+αDmλf(z)-. where Rmf(z) denote the Ruscheweyh derivative, RDmλ f(z) is the generalized Sǎlǎgean operator and An = {f ∈013; H(U), f(z) = z+an+1zn+1+..., z ∈ U} is the class of normalized analytic functions with A1 = A, A certain fuzzy class, denoted by RDFm (δ, λ, α). of analytic functions in the open unit disc is introduced by means of this operator. By making use of the concept of fuzzy differential subordination we will derive various properties and characteristics of the class RDFm (δ, λ, α). Also, several fuzzy differential subordinations are established regarding the operator RDmλ,α. [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]

In the present paper we establish several strong differential subordinations regarding the operator I(m, λ, l), given by I(m, λ, l):Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed, where m ∈ ℕ∪ {0}, λ, l ≥ 0 and A*nζ = {f ∈ H(U x U̅), f(z, ζ) = z + an+1 (ζ) zn+1 +…, z ∈ U, ζ ∈ U̅} is the class of normalized analytic functions. [ABSTRACT FROM AUTHOR]

In the present, paper we establish several strong differential subordinations regardind the new operator Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed, given by Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.: denote the Ruscheweyh derivative, Sn f(z, ζ) is the Sălăgean operator and 휜*nζ = {f ∈ 휢(U x U̅), f(z, ζ) = z + αn+1 (ζ) zn+1 +…, z ∈ U, ζ ∈ U̅} is the class of normalized analytic functions with 휜*1ζ = 휜*ζ. [ABSTRACT FROM AUTHOR]

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]

In the present paper we establish several strong differential subordinations regarding the new operator RIαm,λt, given by RIαm,λt, : A*nζ → A*nζ, RIαm,λtf(z, ζf) = (1 - α)Rmf(z, ζ), where Rmf(z, ζ) denote the Ruscheweyh derivative, I (m, λ, l) is the multiplier transformation and A*nζ = {f ∈ H(U x U̅), f(z, ζ) = z + an+1 (ζ) zn+1 + …, z ∈ U, ζ ∈ U̅} is the class normalized analytic functions. A certain subclass, denoted by RIm (δ, λ, l, α, ζ), of analytic functions is introduced by means of the new operator. [ABSTRACT FROM AUTHOR]

In the present paper we establish several strong differential subordinations regarding the new operator RDnλα, given by RD RDnλα : A*ς → A*ς, RDnλα f (z, ς) = (1 - α) Rn f(z, ς) + α Dn&lambda f(z, ς), where Rn f(z, ς) denote the Ruscheweyh derivative, Dnλ f(z, ς) is the generalized Sălăgean operator and A*nς = {f ∈ H(U x U̅), f(z, ς) = z + an+1 (ς) zn+1 + …, z ∈ U, ς U̅} is the class of normalized analytic functions with A*1ς = A*ς. A certain subclass, denoted by RDn (δ, λ, α), of analytic functions is introduced by means of the new operator. [ABSTRACT FROM AUTHOR]

Abstract: For functions belonging to the class and , of normalized analytic functions in the open unit disc , which is investigated in this paper, the author derives several interesting differential subordination results. These subordinations are established by means of a special case of the multiplier transformations namely where , , and , A number of interesting consequences of some of these subordination 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]

In the present paper we define a new operator using the multiplier transformation and Ruscheweyh derivative. Denote by RIn,λ,lα the operator given by RIn,λ,lα A → A, RIn,λ,lα f (z) = (1-α)Rn f(z)+αI(n, λ, l) f (z); for z ∈ U; where Rn f (z) denote the Ruscheweyh derivative, I (n, λ, l) f (z) is the multiplier transformation and An = {f ∈ H(U) : f (z) = z + an+1zn+1 + …, z ∈ U} is the class of normalized analytic functions with A1 = A. A certain subclass, denoted by RJn (δ, λ, l, α), of analytic functions in the open unit disc is introduced by means of the new operator. By making use of the concept of differential subordination we will derive various properties and characteristics of the class RJn (δ, λ l, α). Also, several differential subordinations are established regardind the operator RIn,λ,lα. [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]

For functions belonging to the class Sm (δ, α), δ ϵ [0, 1), α ≥ 0 and in ϵ N, of normalized analytic functions in the open unit disc U, which are investigated in this paper, the author derives several interesting differential subordination results. These subordinations are established by means of a special case of the multiplier transformations Ip (m, λ, l) ƒ(z) namely >>>>>Ip (m, λ, l) ƒ(z) ≔ zp + Σj=p+n∞ (p + λ(j - 1) + l/p + 1)m ajzj, where p, n ϵ N, m ϵ NU {0}, λ, l ≥ 0 and ƒ ϵ A(p,n), A (p,n) = {ƒ ϵ H(U) : ƒ(z) = zp + Σj=p+n∞ ajzj, z ϵ U). A number of interesting consequences of some of these subordination 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, by means of convolution product between Ruscheweyh operator and the multiplier transformation I (m, λ, l). For functions ƒ belonging to the class A we define the differential operator IRλ,tm : A → A, IRλ,tmƒ (z) := (I (m, λ, l) * Rm) ƒ(z) where An = {ƒ ∈ H(U) ƒ(z) = z + αn+1zn+1 + ߪ, z ࢠ U} is the class of normalized analytic functions with A1 = A. We study certain differential subordinations regarding the operator IRλ,tm. [ABSTRACT FROM AUTHOR]

The present paper deals with notions from the field of complex analysis which have been adapted to fuzzy sets theory, namely, the part dealing with geometric function theory. Several fuzzy differential subordinations are established regarding the operator L α m , given by L α m : A n → A n , L α m f (z) = (1 − α) R m f (z) + α S m f (z) , where A n = { f ∈ H (U) , f (z) = z + a n + 1 z n + 1 + ... , z ∈ U } is the subclass of normalized holomorphic functions and the operators R m f (z) and S m f (z) are Ruscheweyh and Sălăgean differential operator, respectively. Using the operator L α m , a certain fuzzy class of analytic functions denoted by S L F m δ , α is defined in the open unit disc. Interesting results related to this class are obtained using the concept of fuzzy differential subordination. Examples are also given for pointing out applications of the theoretical results contained in the original theorems and corollaries. [ABSTRACT FROM AUTHOR]

Here, we study strong differential subordinations for the extended new operator I R λ , l m defined by the Hadamard product of the extended multiplier transformation I m , λ , l and the extended Ruscheweyh derivative R m , on the class of normalized analytic functions A n ζ ∗ = { f ∈ H (U × U ¯) , f (z , ζ) = z + a n + 1 ζ z n + 1 + ⋯ , z ∈ U , ζ ∈ U ¯ } , by I R λ , l m : A n ζ ∗ → A n ζ ∗ , I R λ , l m f z , ζ = I m , λ , l ∗ R m f z , ζ. [ABSTRACT FROM AUTHOR]