Class Of Admissible Functions Research Articles

Introduction in third-order fuzzy differential subordination

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 José 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. The example given demonstrates the applicability of the new findings.

Differential sandwich subordination of convolution operator acting on complex meromorphic functions

In the present study, a new complex convolution operator is introduced in terms of the meromorphic formula of the generalized Mittag-Leffler functions (MLfs), namely R– function. By employing the subordination methodology, appropriate classes of admissible functions are considered as well as the dual features of the third-order(3rd-order) differential subordination and superordination for this new operator are discussed. As a sequel, the sandwich-type outcome is also examined.

Differential Subordination and Differential Superordination for Classes of Admissible Multivalent Functions Associated with a Linear Operator

In this paper, we first introduce a linear integral operator ℑp(a,c,μ)(μ>0;a,c∈R;c>a>−μp;p∈N+:={1,2,3,…}), which is somewhat related to a rather specialized form of the Riemann–Liouville fractional integral operator and its varied form known as the Erdélyi–Kober fractional integral operator. We then derive some differential subordination and differential superordination results for analytic and multivalent functions in the open unit disk U, which are associated with the above-mentioned linear integral operator ℑp(a,c,μ). The results presented here are obtained by investigating appropriate classes of admissible functions. We also obtain some Sandwich-type results.

Admissible Classes of Multivalent Meromorphic Functions Defined by a Linear Operator

The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on differential subordination, one of the newest techniques used in the field, also known as the technique of admissible functions. For that, the appropriate classes of admissible functions are first defined. Based on these classes, we obtain some differential subordination and superordination results for multivalent meromorphic functions, analytic in the punctured unit disc, related to a linear operator ℑρ,τp(ν), for τ>0,ν,ρ∈C, such that Re(ρ−ν)≧0, Re(ν)>τp,(p∈N). Moreover, taking into account both subordination and superordination results, we derive a sandwich-type theorem. The connection with some other known results and an example are also provided.

Existence and Stability of Smooth Traveling Circular Pairs for the Generalized Surface Quasi-Geostrophic Equation

AbstractIn this paper, we construct smooth traveling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the 2D incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes of admissible functions. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variational setting. Using these facts, we further prove the orbital stability of the circular vortex pairs for the generalized surface quasi-geostrophic equation.

Asymptotics and Uniqueness of Solutions of the Elasticity System with the Mixed Dirichlet–Robin Boundary Conditions

We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at infinity, with the condition that the energy integral with the weight |x|a is finite. Depending on the value of the parameter a, we have proved uniqueness (or non-uniqueness) theorems for the mixed Dirichlet–Robin problem, and also given exact formulas for the dimension of the space of solutions. The main method for studying the problem under consideration is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions.

On The Third-Order Complex Differential Inequalities of ξ-Generalized-Hurwitz–Lerch Zeta Functions

In the z- domain, differential subordination is a complex technique of geometric function theory based on the idea of differential inequality. It has formulas in terms of the first, second and third derivatives. In this study, we introduce some applications of the third-order differential subordination for a newly defined linear operator that includes ξ -Generalized-Hurwitz–Lerch Zeta functions (GHLZF). These outcomes are derived by investigating the appropriate classes of admissible functions.

Identification of the diffusion coefficient in a time fractional diffusion equation

Abstract In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to D t α C ⁢ u - ∇ ⋅ ( k ⁢ ( x ) ⁢ ∇ ⁡ u ) = f {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f} , where the function k = k ⁢ ( x ) {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

Admissible classes of multivalent functions associated with an integral operator

In this paper we investigate some applications of the differential subordination and superordination of classes of admissible functions associated with an integral operator. Additionally, differential sandwich-type results are obtained.

A priori estimates for solutions to a class of obstacle problems under p, q-growth conditions

In this paper we would like to complement the results contained in Gavioli (Forum Math, to appear) by dealing with the higher differentiability of integer order of solutions to a class of obstacle problems under non-standard growth conditions, fulfilling variational inequalities of the kind $$\begin{aligned} \int _{\varOmega } \langle {\mathcal {A}}(x, Du), D(\varphi - u) \rangle \, dx \ge 0 \qquad \forall \, \varphi \in {\mathcal {K}}_{\psi }(\varOmega ). \end{aligned}$$Here the operator $${\mathcal {A}}$$ satisfies p, q-growth conditions with p and q related by 1$$\begin{aligned} \frac{q}{p} n$$. More precisely the function $$\psi \in W^{1,p}(\varOmega )$$, called obstacle, is such that $$D\psi \in W^{1,r}_{\mathrm{loc}}(\varOmega )$$ and $${\mathcal {K}}_{\psi }=\{w \in W^{1,p}(\varOmega ): w \ge \psi \,\, \text {a.e. in }\varOmega \}$$ is the class of admissible functions. The main difference with the previous work (Gavioli in Forum Math, to appear) is that here we assume the same regularity both for the gradient of the obstacle $$D\psi$$ and for the partial map $$x\mapsto {\mathcal {A}}(x,\xi )$$, that is, a higher differentiability of Sobolev order in the space $$W^{1,r}$$ with the same $$r>n$$ appearing in (1). For the sake of clarity, we focus on the derivation of the a priori estimates since the approximation procedure is standard and can be found in Cupini et al. (Nonlinear Anal 154:7–24, 2017), Cupini et al. (Differ Equ 265(9):4375–4416, 2018), Cupini et al. (Nonlinear Anal 54(4):591–616, 2003), Eleuteri et al. Ann Mat Pura Appl (195(5):1575–1603, 2016) and Gavioli (Forum Math, to appear).

Applications of differential subordinations involving a generalized fractional differintegral operator

Using the third-order differential subordination basic results, we introduce certain classes of admissible functions and investigate some applications of third-order differential subordination for p-valent functions associated with generalized fractional differintegral operator.

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

Abstract The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k-fold Lebesgue measure on ℝ n in terms of the corresponding measures on k-dimensional linear subspaces of ℝ n . We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayley-Laplace operator on matrix spaces. Another application of our proof is the celebrated Drury identity that plays a key role in the study of mapping properties of the Radon-John k-plane transforms. Our proof gives precise meaning to the constants in Drury’s identity and to the class of admissible functions.

Admissible classes of multivalent functions associated with Liu-Owa integral operator

In this paper we investigate some applications of second order differential subordination and superordination for classes of admissible functions associated with Liu-Owa integral operator. Additionally, differential sandwich-type results are obtained.

Quasi Solution of a Nonlinear Inverse Parabolic Problem

In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to $$ \aleph (u):\equiv u_{t}-\nabla \cdot (F(x,\nabla u)) $$ where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function F. At the first step of the methodology, we give a stability result corresponding to connectivity of F and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions.

Differential subordination associated with generalized Mittag-Leffler function

A new operator \(\mathcal {E}^\gamma _{\alpha ,\beta }f(z)\) associated with the generalized Mittag-Leffler function \(E^\gamma _{\alpha ,\beta }(z)\) is defined in the unit disk. Second-order differential subordination results for operator \(\mathcal {E}^\gamma _{\alpha ,\beta }f(z)\) are obtained by investigating appropriate classes of admissible functions. Inequalities for normalized Mitttag-Leffler function \(\mathbb {E}^\gamma _{\alpha ,\beta }(z)\) and hyperbolic functions are also obtained.

An integrodifferential operator for meromorphic functions associated with the Hurwitz-Lerch zeta function

In this paper, we introduce a new integrodifferential operator associated with the Hurwitz Lerch Zeta function in the puncture open disk of the meromorphic functions. We also obtain some properties of the third-order differential subordination and superordination for this integrodifferential operator, by using certain classes of admissible functions.

STRONG DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF NEW GENERALIZED DERIVATIVE OPERATOR

In this work, certain classes of admissible functions are considered. Some strong differential subordination and superordination properties of analytic functions associated with new generalized derivative operator $\mathfrak{B}^{\mu,q,s}_{\lambda_{1},\lambda_{2},\ell,d}$ are investigated. New strong differential sandwich-type results associated with the generalized derivative operator are also given.

Differential Subordinations and Superordinations of Certain Meromorphic Functions associated with an Integral Operator

Difierential subordinations and superordinations results are obtained for cer- tain meromorphic functions in the punctured unit disk which are associated with an inte- gral operator. These results are obtained by investigating appropriate classes of a dmissible functions. Sandwich-type results are also obtained.

Strong Differential Subordination and Superordination for Multivalently Meromorphic Functions Involving the Hurwitz–Lerch Zeta Function

An integral operator for meromorphic multivalent function in the punctured unit disk is introduced by means of Hurwitz–Lerch Zeta function. Strong differential subordination and superordination results associated with the operator are obtained by investigating appropriate classes of admissible functions.

Third-order differential subordination and superordination involving a fractional operator

Abstract The third-order differential subordination and the corresponding differential superordination problems for a new linear operator convoluted the fractional integral operator with the Carlson-Shaffer operator, are investigated in this study. The new operator satisfies the required first-order differential recurrence (identity) relation. This property employs the subordination and superordination methodology. Some classes of admissible functions are determined, and these significant classes are exploited to obtain fractional differential subordination and superordination results. The new third-order differential sandwich-type outcomes are investigated in subsequent research.