Refinements of the Converse Hölder and Minkowski Inequalities

We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the converse Minkowski inequality for functionals and of its continuous form and give refinements of it. Application on integral mixed means is given.

Refinements of Jensen's inequality and applications

<abstract><p>The principal aim of this research work is to establish refinements of the integral Jensen's inequality. For the intended refinements, we mainly use the notion of convexity and the concept of majorization. We derive some inequalities for power and quasi–arithmetic means while utilizing the main results. Moreover, we acquire several refinements of Hölder inequality and also an improvement of Hermite–Hadamard inequality as consequences of obtained results. Furthermore, we secure several applications of the acquired results in information theory, which consist bounds for Shannon entropy, different divergences, Bhattacharyya coefficient, triangular discrimination and various distances.</p></abstract>

Determination of Bounds for the Jensen Gap and Its Applications

The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.

Higher integrability and reverse Hölder inequalities in the limit cases

We study the higher integrability of weights satisfying a reverse Hölder inequality ( ⨏ I u β ) 1 β ≤ B ⁢ ( ⨏ I u α ) 1 α {\biggl{(}\fint_{I}u^{\beta}\biggr{)}^{\frac{1}{\beta}}}\leq B{\biggl{(}\fint_{I}u^{\alpha}\biggr{)}^{\frac{1}{\alpha}}} for some B > 1 B>1 and given α < β \alpha<\beta , in the limit cases when α ∈ { - ∞ , 0 } \alpha\in\{-\infty,0\} and/or β ∈ { 0 , + ∞ } \beta\in\{0,+\infty\} . The results apply to the Gehring and Muckenhoupt weights and their corresponding limit classes.

Characterization of the non-homogenous Dirac-harmonic equation

We introduce the non-homogeneous Dirac-harmonic equation for differential forms and characterize the basic properties of solutions to this new type of differential equations, including the norm estimates and the convergency of sequences of the solutions. As applications, we prove the existence and uniqueness of the solutions to a special non-homogeneous Dirac-harmonic equation and its corresponding reverse Hölder inequality.

The reverse Holder inequality for an elementary function

For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by \smallskip\centerline{$\displaystyle\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\qquad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$} Assume that $0<A<B$, $0<\theta<1$ and consider the step function$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$, where $\chi_E$ is the characteristic function of the set $E$. Let $-\infty<p<q<+\infty$. The main result of this work consists in finding the term \smallskip\centerline{$\displaystyleC_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$} \smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\theta_{p<q,A<B}$ with respect to $\beta=B/A\in(1,+\infty)$.The cases $p=0$ or $q=0$ are considered separately. The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].

Stochastic version of Henry type Gronwall’s inequality

We use Young’s and Hölder inequality combined with classical Gronwall’s inequality to derive present a new version of the stochastic of Gronwall’s inequalities with singular kernels.

Inequalities on the Generalized ABC Index

In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for ABCα improved, when α=1/2, known results on the ABC index. Moreover, in order to obtain our results, we proved a kind of converse Hölder inequality, which is interesting on its own.

Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral

The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically ψ - s -convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically ψ - s -convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.