Differential superordination for harmonic complex-valued functions

Georgia Irina Oros;  ; Gheorghe Oros;  ;

doi:10.24193/subbmath.2019.4.04

Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions

Mathematics ◽

10.3390/math8112041 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2041

Author(s):

Georgia Irina Oros

Keyword(s):

Analytic Functions ◽

Dual Problem ◽

Differential Subordination ◽

Research Subject ◽

Harmonic Complex ◽

Differential Superordination ◽

Complex Valued ◽

Differential Subordinations

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.

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A New Approach to Non-Singular Plane Cracks Theory in Gradient Elasticity

Mathematical and Computational Applications ◽

10.3390/mca24040093 ◽

2019 ◽

Vol 24 (4) ◽

pp. 93

Author(s):

Lurie ◽

Volkov-Bogorodsky ◽

Vasiliev

Keyword(s):

Fracture Mechanics ◽

Helmholtz Equation ◽

Scale Parameter ◽

Crack Tip ◽

Theory Of Elasticity ◽

Singular Problem ◽

Harmonic Complex ◽

Concentration Problem ◽

Non Local ◽

Complex Valued

A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right side of the Helmholtz equation is a singular classical solution. An assertion is proved that allows us to propose a new solution for displacements and stresses at the crack tip through the vector harmonic potential, which determines by the Papkovich–Neuber representation. One of the goals of this work is a definition of a new representation of the solution of the plane problem of the theory of elasticity through the complex-valued harmonic potentials included in the Papkovich–Neuber relations represented in a symmetric form, which is convenient for applications. It is shown here that this new representation of the solution for the mechanics of cracks can be written through one harmonic complex-valued potential. The explicit potential value is found by comparing the new solution with the classical representation of the singular solution at the crack tip constructed using the complex potentials of Kolosov–Muskhelishvili. A generalized solution of the singular problem of fracture mechanics is reduced to a non-singular stress concentration problem, which allows one to implement a new concept of non-singular fracture mechanics, where the scale parameter along with ultimate stresses determines the fracture criterion and is determined by experiments.

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Differential Subordination and Superordination Results Using Fractional Integral of Confluent Hypergeometric Function

Symmetry ◽

10.3390/sym13020327 ◽

2021 ◽

Vol 13 (2) ◽

pp. 327

Author(s):

Alina Alb Lupaş ◽

Georgia Irina Oros

Keyword(s):

Hypergeometric Function ◽

Fractional Integral ◽

Type Theorem ◽

Confluent Hypergeometric Function ◽

Differential Subordination ◽

Different Types ◽

Sandwich Type ◽

Differential Superordination ◽

Differential Subordination And Superordination ◽

Complex Valued

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.

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