Briot–Bouquet Differential Subordinations for Analytic Functions Involving the Struve Function

In this paper, it is aimed to determine the radii of starlikeness and convexity of the normalized generalized Struve functions for three different kinds of normalization and to find tight lower and upper bounds for the radius of starlikeness and convexity of these normalized Struve functions by making use of Euler-Rayleigh inequalities. The Laguerre-Polya class of entire functions has a crucial role in constructing our main results. *********************************************************************

This project aimed to gather together the latest developments in research concerning complex-valued functions from the perspective of geometric function theory. Scholars contributions weresought on topics including, but not limited to: new classes of univalent and bi-univalent functions; studies regarding coefficient estimates including the FeketeSzego functional, Hankel determinants, and Toeplitz matrices; applications of different types of operators in geometric function theory including differential, integral, fractional, or quantum calculus operators; differential subordination and superordination theories in their classical form and also concerning their recent extensionsstrong and fuzzy differential subordination and superordination theories; applications of different hypergeometric functions and orthogonal polynomials in geometric function theory. The presentation of new results obtained by using any other techniques which can be applied in the field of complex analysis were also welcomed. Hopefully, through this project, new lines of research associatedwithgeometric function theory have been highlighted and willserve toboost development inthis field.

<abstract><p>This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order $ \lambda $ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular.</p></abstract>

Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators.

A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

In light of the well-established and widely-used theory of differential subordination, recent works incorporating fuzzy elements into Geometric Function Theory have given rise to the concept of fuzzy differential subordination. Second-order fuzzy differential subordinations were taken into consideration for studies up until this point. The research described in this paper aims to expand the concept of fuzzy differential subordination to third-order fuzzy differential subordination, building on an idea first put forth in 2011 by Jos\'{e} A. Antonino and Sanford S. Miller and still being investigated by scholars today. The key concepts and preliminary findings required for the development of this branch of fuzzy differential subordination are introduced. The class of admissible functions is specified, the fundamental theorems are established and the fundamental concepts of the third-order fuzzy subordination approach are presented. Several examples constructed as applications of the new results demonstrate the applicability of the new findings.

In this paper, we derive some subordination and superordination results for the generalized "Srivastava-Attiya" fractional integral operator. Some interesting corollaries for this operator is also obtained.

Background The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) .

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function.

In this paper we introduce and examine the differential subordination related to the geometric mean of the form [p(z)]1-γp(z)+zp′(z)φp(z),zp′(z)γ≺h(z),z∈D,where D:={z∈C:|z|<1} and h is a convex univalent function with 0∈∂h(D). In particular, the above differential subordination generalizes the well-known Briot-Bouquet differential subordination. At the same time new type of ordinary differential equation has been proposed for study. The main results allow us to construct non-obvious subclasses in the class of holomorphic functions on D.

A method of calculating the Green’s function for the Laplace equation is suggested. This method is based on the use of properties of the special Bessel and Struve functions. An example of calculations is given.

The paper focuses on the calculation of mutual inductance between underground inducing line (underground power cable) and underground induced line (underground telecommunication cable). The calculation for the infinite integrals in the expression was simplified. The infinite integrals was converted to Watson infinite integrals by expanding the exponential function in series, so the known higher-order Neumann function and Struve function could be used to express the Watson infinite function, thus the numerical calculation problem for the mutual inductance between the underground power cable and the underground telecommunication cable is solved. The approximate expression which was more convenient to calculate could also be obtained, if the higher-order Neumann function, the first and the second item of expanded Struve function is fetched.

The main object of this paper is to find necessary and sufficient conditions for generalized Struve functions of first kind to be in the classes SPp(α,β) and UCSP(α,β) of uniformly spirallike functions and also give necessary and sufficient conditions for z(2 − up(z)) to be in the above classes. Furthermore, we give conditions for the integral operator L(m,c,z) = \int_{0}^{z} (2 − up(t))dt to be in the class UCSPT(α,β). Several corollaries and consequences of the main results are also considered.

In this article, the Saigo's k-fractional order integral and derivative operators involving k-hypergeometric function in the kernel are applied to the k-Struve function; outcome are expressed in the term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Struve functions are considered.

The idea of fuzzy differential subordination is a generalisation of the traditional idea of differential subordination that evolved in recent years as a result of incorporating the idea of fuzzy set into the field of geometric function theory. In this investigation, we define some general classes of p-valent analytic functions defined by the fuzzy subordination and generalizes the various classical results of the multivalent functions. Our main focus is to define fuzzy multivalent functions and discuss some interesting inclusion results and various other useful properties of some subclasses of fuzzy p-valent functions, which are defined here by means of a certain generalized Srivastava-Attiya operator. Additionally, links between the significant findings of this study and preceding ones are also pointed out.