Applications of Differential Subordination for Argument Estimates of Multivalent Analytic Functions

Research Subject ◽

Harmonic Complex ◽

Differential Superordination ◽

Complex Valued ◽

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.

Differential Subordinations for Nonanalytic Functions

Abstract and Applied Analysis ◽

10.1155/2014/251265 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Georgia Irina Oros ◽

Gheorghe Oros

Keyword(s):

Complex Plane ◽

Classical Theory ◽

Analytic Functions ◽

Unit Disc ◽

Sufficient Conditions ◽

Analytic Case ◽

Complex Functions ◽

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classesC1(U), respectively, andC2(U)to be univalent and to mapUonto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the classC1which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classesC1andC2following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). LetΩbe any set in the complex planeC, letpbe a nonanalytic function in the unit discU,p∈C2(U),and letψ(r,s,t;z):C3×U→C. In this paper, we consider the problem of determining properties of the functionp, nonanalytic in the unit discU, such thatpsatisfies the differential subordinationψ(p(z),Dp(z),D2p(z)-Dp(z);z)⊂Ω⇒p(U)⊂Δ.

Strong Differential Superordination Results Involving Extended Sălăgean and Ruscheweyh Operators

Mathematics ◽

10.3390/math9192487 ◽

2021 ◽

Vol 9 (19) ◽

pp. 2487

Author(s):

Alina Alb Lupaş ◽

Georgia Oros

Keyword(s):

Analytic Functions ◽

Holomorphic Functions ◽

New Approach ◽

Ruscheweyh Derivative ◽

Differential Superordination ◽

Strong Differential Subordination ◽

Extended Operator ◽

Dual Notion ◽

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.

On a certain subclass of analytic functions defined by a generalized S˘al˘agean operator and Ruscheweyh derivative

Carpathian Journal of Mathematics ◽

10.37193/cjm.2012.02.20 ◽

2012 ◽

Vol 28 (2) ◽

pp. 183-190

Author(s):

ALINA ALB LUPAS ◽

Keyword(s):

Analytic Functions ◽

Unit Disc ◽

Open Unit ◽

Open Unit Disc ◽

Ruscheweyh Derivative ◽

In the present paper we define a new operator using the generalized Sal˘ agean and Ruscheweyh operators. Denote by ˘ RDm λ,α the operator given by RDm λ,α : An → An, RDm λ,αf(z) = (1 − α)Rmf(z) + αDm λ f(z), z ∈ U, where Rmf(z) denote the Ruscheweyh derivative, Dm λ f(z) is the generalized Sal˘ agean operator and ˘ An = {f ∈ H(U) : f(z) = z +an+1z n+1 +. . . , z ∈ U} is the class of normalized analytic functions. A certain subclass, denoted by RDm (δ, λ, 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 RDm (δ, λ, α) . Also, several differential subordinations are established regarding the operator RDm λ,α.

Some Properties for Strong Differential Subordination of Analytic Functions Involving Ruscheweyh Derivative Operator

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.4120.6370 ◽

2020 ◽

pp. 63-70

Author(s):

Asraa Abdul Jaleel Husien

Keyword(s):

Unit Disk ◽

Analytic Functions ◽

Open Unit ◽

Derivative Operator ◽

Closed Unit Disk ◽

Ruscheweyh Derivative Operator ◽

Ruscheweyh Derivative ◽

Strong Differential Subordination ◽

In this paper, we introduce and study some properties for strong differential subordinations of analytic functions associated with Ruscheweyh derivative operator defined in the open unit disk and closed unit disk of the complex plane.

Some Properties for Strong Differential Subordination of Analytic Functions Associated with Wanas Operator

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.4120.2938 ◽

2020 ◽

pp. 29-38

Author(s):

Abbas Kareem Wanas ◽

Pall-Szabo Agnes Orsolya

Keyword(s):

Unit Disk ◽

Complex Plane ◽

Analytic Functions ◽

Open Unit ◽

Open Unit Disk ◽

Closed Unit Disk ◽

Strong Differential Subordination ◽

In this paper, by making use of Wanas operator, we derive some properties related to the strong differential subordinations of analytic functions defined in the open unit disk and closed unit disk of the complex plane.

On Special Differential Subordinations Using Fractional Integral of Sălăgean and Ruscheweyh Operators

Symmetry ◽

10.3390/sym13091553 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1553

Author(s):

Alina Alb Lupaş ◽

Georgia Irina Oros

Keyword(s):

Analytic Functions ◽

Unit Disc ◽

Fractional Integral ◽

In the present paper, a new operator denoted by Dz−λLαn is defined by using the fractional integral of Sălăgean and Ruscheweyh operators. By means of the newly obtained operator, the subclass Snδ,α,λ of analytic functions in the unit disc is introduced, and various properties and characteristics of this class are derived by applying techniques specific to the differential subordination concept. By studying the operator Dz−λLαn, some interesting differential subordinations are also given.

Applications of third order differential subordination and superordination involving generalized struve function

Filomat ◽

10.2298/fil1910047g ◽

2019 ◽

Vol 33 (10) ◽

pp. 3047-3059 ◽

Cited By ~ 1

Author(s):

Priyabrat Gochhayat ◽

Anuja Prajapati

Keyword(s):

Linear Operator ◽

Analytic Functions ◽

Third Order ◽

The Third ◽

Sandwich Type ◽

Differential Subordination And Superordination ◽

Struve Functions ◽

Differential Subordinations ◽

Admissible Functions

In the present paper, by making use of the linear operator associated with generalized Struve functions suitable classes of admissible functions are investigated and the dual properties of the third-order differential subordinations are presented. As a consequence, various sandwich-type results are established for a class of univalent analytic functions involving generalized Struve functions. Relevant connections of the new results presented here with those that were considered in earlier works are pointed out.

On Certain Classes ofp-Valent Functions by Using Complex-Order and Differential Subordination

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/275935 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12

Author(s):

Abdolreza Tehranchi ◽

Adem Kılıçman

Keyword(s):

Integral Operator ◽

Unit Disk ◽

Complex Number ◽

Analytic Functions ◽

Complex Order ◽

Related Integral ◽

The aim of the present paper is to study thep-valent analytic functions in the unit disk and satisfy the differential subordinationsz(Ip(r,λ)f(z))(j+1)/(p-j)(Ip(r,λ)f(z))(j)≺(a+(aB+(A-B)β)z)/a(1+Bz),whereIp(r,λ)is an operator defined by Sălăgean andβis a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties.

On differential subordinations for a class of analytic functions defined by a linear operator

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204309208 ◽

2004 ◽

Vol 2004 (42) ◽

pp. 2219-2230

Author(s):

V. Ravichandran ◽

Herb Silverman ◽

S. Sivaprasad Kumar ◽

K. G. Subramanian

Keyword(s):

Linear Operator ◽

Analytic Functions ◽

Linear Operators ◽

Special Cases ◽

We obtain several results concerning the differential subordination between analytic functions and a linear operator defined for a certain family of analytic functions which are introduced here by means of these linear operators. Also, some special cases are considered.