Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

We consider functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding real n-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings

Combinatorial extensions of Popoviciu\u27s inequality via Abel-Gontscharoff polynomial with applications in information theory

We establish new refinements and improvements of Popoviciu’s inequality for n-convex functions using Abel-Gontscharoff interpolating polynomial along with the aid of new Green functions. We construct new inequalities for n-convex functions and compute new upper bounds for Ostrowski and Grüss type inequalities. As an application of our work in information theory, we give new estimations for Shannon, Relative and Zipf-Mandelbrot entropies using generalized Popoviciu’s inequality

Weighted Popoviciu type inequalities via generalized Montgomery identities

We obtained useful identities via generalized Montgomery identities, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the Čebyšev functional. Some results relating to the Grüss and Ostrowski type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functionals associated with the obtained inequalities

Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

We consider functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding real n-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings

NON-SYMMETRIC STOLARSKY MEANS

Abstract. In this paper we construct n -exponentially convex functions and exponentially convex functions using the functional defined as the difference of the right parts of the HermiteHadamard inequality, for different classes of functions. Applying these results on some starshaped functions, we derive non-symmetric means of Stolarsky type

Superquadratic stochastic processes and their fractional perspective with applications in information theory

Superquadraticity is a generalization of convexity that yields more refined results compared to those obtained through convexity alone. In this work, we established, for the first time, a class of superquadratic stochastic processes and explored their fundamental properties. Based on these properties, we derived Jensen's and (Hermite-Hadamard) $\mathbb{HH}$'s type inequalities, along with their fractional counterparts, in the context of mean-square stochastic (Riemann-Liouville) $\mathbb{R.L}$ fractional integrals. The validity of our findings was supported by graphical illustrations using suitable examples. Furthermore, we extended the applicability of our results to information theory by introducing several stochastic divergence measures

Construction of new fractional inequalities via generalized n-fractional polynomial s-type convexity

This paper focuses on introducing and investigating the class of generalized $n$-fractional polynomial $s$-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowski-type inequalities are presented for generalized $n$-fractional polynomial $s$-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized $n$-fractional polynomial $s$-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the Riemann-Liouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields

Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re$, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $\mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re$. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $\mathrm{b_{1}}$, and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus

Pain after upper limb surgery under peripheral nerve block is associated with gut microbiome composition and diversity

peer-reviewedGut microbiota play a role in certain pain states. Hence, these microbiota also influence somatic pain. We aimed to determine if there was an association between gut microbiota (composition and diversity) and postoperative pain. Patients (n = 20) undergoing surgical fixation of distal radius fracture under axillary brachial plexus block were studied. Gut microbiota diversity and abundance were analysed for association with: (i) a verbal pain rating scale of < 4/10 throughout the first 24 h after surgery (ii) a level of pain deemed “acceptable” by the patient during the first 24 h following surgery (iii) a maximum self-reported pain score during the first 24 h postoperatively and (iv) analgesic consumption during the first postoperative week. Analgesic consumption was inversely correlated with the Shannon index of alpha diversity. There were also significant differences, at the genus level (including Lachnospira), with respect to pain being “not acceptable” at 24 h postoperatively. Porphyromonas was more abundant in the group reporting an acceptable pain level at 24 h. An inverse correlation was noted between abundance of Collinsella and maximum self-reported pain score with movement. We have demonstrated for the first time that postoperative pain is associated with gut microbiota composition and diversity. Further work on the relationship between the gut microbiome and somatic pain may offer new therapeutic targets