Narrow Results

In this paper, we propose a new fractional operator which is based on the weight function for Atangana–Baleanu $({𝒜}{ℬ})$-fractional operators. A motivating characteristic is the generalization of classical variants within the weighted ${𝒜}{ℬ}$-fractional integral. We aim to establish Minkowski and reverse Hölder inequalities by employing weighted ${𝒜}{ℬ}$-fractional integral. The consequences demonstrate that the obtained technique is well-organized and appropriate.

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete $\hslash$-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete $\hslash$-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla $\hslash$ fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

Discrete fractional calculus (${𝒟}{ℱ}{𝒞}$) is significant for neural networks, complex dynamic systems and frequency response analysis approaches. In contrast with the continuous-time frameworks, fewer outcomes are accessible for discrete fractional operators. This study investigates some major consequences of two sorts of inequalities by considering discrete Atangana–Baleanu $({𝒜}{ℬ})$-fractional operator having $\hslash$-discrete generalized Mittag-Leffler kernels in the sense of Riemann type (${𝒜}{ℬ}{ℛ}$). Certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete ${𝒜}{ℬ}$-fractional operators having $\hslash$-discrete generalized Mittag-Leffler kernels are given. Moreover, several other generalizations can be generated for nabla $\hslash$-fractional sums. The proposing discretization is a novel form of the existing operators that can be provoked by some intriguing features of chaotic systems to design efficient dynamics description in short time domains. Furthermore, by combining two mechanisms, numerous new special cases are introduced.

We prove a very general sharp inequality of the Hölder–Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong–Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson’s hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.