A novel algorithmic multi-attribute decision-making framework for solar panel selection using modified aggregations of cubic intuitionistic fuzzy hypersoft set.

To address the shortcomings of the cubic intuitionistic fuzzy sets (CIFSs) for the entitlement of multi-argument approximate function, the cubic intuitionistic fuzzy hypersoft set (Ω-set) is an emerging study area. This type of setting associates the sub-parametric tuples with the collection of CIFSs. Categorizing the evaluation of parameters into their corresponding sub-parametric values based on non-overlapping sets has significance in decision making and optimization related situations. Some operations of Ω-set are proposed in this study, along with certain practical features. We provide the complement, P-order, and R-order subsets, P-union ( ∪ P ), R-union ( ∪ R ), P-intersection ( ∩ P ) and R-intersection ( ∩ R ) of Ω-sets. The internal cubic intuitionistic fuzzy hypersoft set ( Ω I -set) and the external cubic intuitionistic fuzzy hypersoft set ( Ω E -set) are also proposed in this paper, which will aid researchers in applying this new theory to other areas of study. We show a few examples in this context and look into some more aspects of ∪ P , ∪ R , ∩ P and ∩ R of Ω I -sets and Ω E -sets. Arguments for a few significant theorems about Ω I -sets and Ω E -sets are also presented. Lastly, an algorithm is presented that assists decision-makers in evaluating appropriate solar panels to establish solar plants. The proposed algorithm uses the idea of ∪ P and ∪ R for two Ω-sets constructed based on expert opinions of decision makers.

Refinement of Jensen's inequality and estimation of f- and Rényi divergence via Montgomery identity.

Jensen's inequality is important for obtaining inequalities for divergence between probability distribution. By applying a refinement of Jensen's inequality (Horváth et al. in Math. Inequal. Appl. 14:777-791, 2011) and introducing a new functional based on an f-divergence functional, we obtain some estimates for the new functionals, the f-divergence, and Rényi divergence. Some inequalities for Rényi and Shannon estimates are constructed. The Zipf-Mandelbrot law is used to illustrate the result. In addition, we generalize the refinement of Jensen's inequality and new inequalities of Rényi Shannon entropies for an m-convex function using the Montgomery identity. It is also given that the maximization of Shannon entropy is a transition from the Zipf-Mandelbrot law to a hybrid Zipf-Mandelbrot law.

On generalization of refinement of Jensen's inequality using Fink's identity and Abel-Gontscharoff Green function.

In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the 'two-point right focal' problem. We use Fink's identity and a new Abel-Gontscharoff-type Green's function for a 'two-point right focal' to generalize the refinement of Jensen's inequality given in (Horváth and Pecaric in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen's inequality using inequalities for the Cebysev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed.

Generalizations of some fractional integral inequalities via generalized Mittag-Leffler function.

Fractional inequalities are useful in establishing the uniqueness of solution for partial differential equations of fractional order. Also they provide upper and lower bounds for solutions of fractional boundary value problems. In this paper we obtain some general integral inequalities containing generalized Mittag-Leffler function and some already known integral inequalities have been produced as special cases.