Holder Inequality Research Articles

On the generalization of Hermite-Hadamard type inequalities for [formula omitted]-convex function via fractional integrals

The main motivation in this article is to prove new integral identities and related results. In this paper, we deal with E`-convex function, Hermite-Hadamard type inequalities, and Katugampola fractional integrals. Firstly, we established the Hermite-Hadamard type inequalities for E`-convex function via Katugampola fractional integrals operators. After that we obtain several new identities and give new results of Holder's inequality, Jensen's integral inequality, Young's inequality, and Power mean inequality for E`-convex function by using Katugampola fractional integral operators. We make connections of our results with various results recognized in the literature.

On estimation of values of jumps for discrete distribution functions

We obtain new inequalities for values of jumps for discrete distribution functions. Values of jumps are bounded by linear combinations of a finite number of moments of the distributions. Obtained inequalities can be used for estimation of ranges for values of improbable jumps when frequencies are zero and are not interesting as estimators. We discuss relationships of derived inequalities with inequalities for probabilities of unions of events and the Caushy-Bunyakovski and Holder inequalities as well.

On conformable fractional versions of trapezoid-type inequalities according to twice-differentiable functions

This paper proves an equality for the case of twice-differentiable convex mappings with respect to the conformable fractional integrals. With the help of this equality, several trapezoid-type inequalities are established by convex functions involving conformable fractional integrals. Sundry significant inequalities are acquired with taking advantage of the convexity, the Holder inequality, and the ¨ power mean inequality. Furthermore, we present several new results connected with trapezoid-type inequalities by using the special choices of obtained results.

Improvement in Some Inequalities via Jensen–Mercer Inequality and Fractional Extended Riemann–Liouville Integrals

The primary intent of this study is to establish some important inequalities of the Hermite–Hadamard, trapezoid, and midpoint types under fractional extended Riemann–Liouville integrals (FERLIs). The proofs are constructed using the renowned Jensen–Mercer, power-mean, and Holder inequalities. Various equalities for the FERLIs and convex functions are construed to be the mainstay for finding new results. Some connections between our main findings and previous research on Riemann–Liouville fractional integrals and FERLIs are also discussed. Moreover, a number of examples are featured, with graphical representations to illustrate and validate the accuracy of the new findings.

K-fractional Ostrowski type inequalities via (s, r)-conex

We are introducing very first time a generalized class named it the class of (s, r)-convex in mixed kind, this class includes s-convex in 1st and 2nd kind, P-convex, quasi-convex, and the class of ordinary convex. Also, we would like to state the generalization of the classical Ostrowski inequality via k-fractional integrals, which is obtained for functions whose first derivative in absolute values is (s, r)-convex in mixed kind. Moreover, we establish some Ostrowski type inequalities via k-fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are (s, r)-convex in mixed kind by using different techniques including Holder's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means would also be given.

Specht's ratio and logarithmic mean in time scale dynamic inequalities and their retrospective variants

In this research article, we investigate reverse Radon's inequality, reverse Bergström's inequality, the reverse weighted power mean inequality, reverse Schlömilch's inequality, reverse Bernoulli's inequality and reverse Lyapunov's inequality with Specht's ratio on time scales. We also present reverse Rogers--Holder's inequality with logarithmic mean and Specht's ratio on time scales. The time scale dynamic inequalities unify and extend some continuous inequalities and their corresponding discrete and quantum versions.

Stochastic fractional differential equations with generalized Caputo's derivative and impulsive effects

In this paper, impulsive stochastic fractional differential equations (ISFDEs) in Lp (p> 2) space are introduced. We present a general framework for finding solution for ISFDEs. Then, by using the Burkholder - Davis - Gundy inequality and Holder's inequality, we prove the existence and uniqueness of solution to ISFDE by fixed point theorem. We also investigate Lipschitz continuity of solutions with respect to initial values by using Gronwall inequality. Finally, we provide an application to illustrate the results we obtained.

Optimal control for Hilfer fractional neutral integrodifferential evolution equations with infinite delay

AbstractThe optimal control solutions for Hilfer fractional neutral integrodifferential evolution equations with infinite delay in Banach space are investigated in this study. To begin, we use fractional computations, Holder inequality, and Banach fixed point approaches to examine conclusions for fractional integrodifferential evolution equations. Following that, we created the fractional integrodifferential evolution system's continuous dependency. We also looked at optimal controls for the Lagrange problem. Finally, an application is given to extract the primary theory.

On Some Generalized Number Theoretic Functions and Ighachane-Akkouchi Holder's Inequalities

Aims/ Objectives: Recently, it has been shown by Ighachane and Akkouchi [1] that using binomial coefficients, one can derive some new refinements of Holder’s inequalities. This inequalities then can be applied to a wide class of special functions such as the Nielsen’s beta function and some extended gamma functions. In this paper, we have derived some generalizations of previously known number theoretic functions. Furthermore, based on the results of Ighachane and Akkouchi, Holder’s inequalities for the derived generalized functions are established.

Fuzzy Ostrowski type inequalities via ϕ - λ -convex functions

We would like to state well-known Ostrowski inequality via \(\phi\)-\(\lambda\)-convex by using the Fuzzy Reimann integrals. In addition, we establish some Fuzzy Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are \(\phi\)-\(\lambda\)-convex by Holder's and power mean inequalities. We are introducing very first time that the class of \(\phi\)-\(\lambda\)-convex function, which is the generalization of many important classes including class of \(h\)-convex, Godunova-Levin \(s\)-convex, \(s\)-convex in the \(2^{\rm nd}\) kind and hence contains convex functions. It also contains class of \(P\)-convex and class of Godunova-Levin. In this way we also capture the results with respect to convexity of functions.

Complement to the Holder Inequality for Multiple Integrals: I

Complement to the Holder Inequality for Multiple Integrals: I

Robust secure beamforming algorithm for intelligent reflecting surface-assisted satellite-terrestrial integrated networks

For intelligent reflection surface (IRS) assisted satellite-terrestrial integrated network (STIN), a robust secure beamforming method based on the imperfect channel state information (CSI) of eavesdroppers is proposed. First of all, considering that the satellite uses spot beam technology to serve the earth station, while the ground base station (BS) serves multiple ground users through multicast technology, and in the case of frequency spectrum shared between the two networks, a joint optimization problem is formulated to minimize the total transmit power of STIN while satisfying the quality of service constraints of the BS users and the achievable secrecy rate constraints of the earth station. Secondly, in order to solve this non-convex problem, the upper bound and the lower bound of the eavesdroppers’ output signal-to-noise ratio under the condition of imperfect CSI are derived by employing triangle inequality together with Holder inequality. Then, a joint optimization scheme of robust BF and power control based on semi-positive definite programming and penalty function method is proposed to realize the secure and reliable transmission in STIN. Finally, the simulation results verify the effectiveness and superiority of the proposed algorithm.

The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator

<abstract><p>The theory of convex function has a lot of applications in the field of applied mathematics and engineering. The Caputo-Fabrizio non-singular operator is the most significant operator of fractional theory which permits to generalize the classical theory of differentiation. This study consider the well known Hermite-Hadamard type and associated inequalities to generalize further. To full fill this mileage, we use the exponential convexity and fractional-order differential operator and also apply some existing inequalities like Holder, power mean, and Holder-Iscan type inequalities for further extension. The generalized exponential type fractional integral Hermite-Hadamard type inequalities establish involving the global integral. The applications of the developed results are displayed to verify the applicability. The establish results of this paper can be considered an extension and generalization of the existing results of convex function and inequality in literature and we hope that will be more helpful for the researcher in future work.</p></abstract>

Complement to the Holder inequality for multiple integrals. II

This article is the second and final part of the author’s work published in the previous issue of the journal. The main result of the article is the statement that if for functions (...) , where m (...) 2 and the numbers p1,...,pm ∈ (1, +∞] are such that 1/p1 + ... + 1/pm < 1 the “non-resonant” condition is fulfilled (the concept introduced by the author in the previous work for functions from spaces L^p(R^n), p ∈ (1, +∞]), then: (...), where [a, b] - n-dimensional parallelepiped, the constant C > 0 does not depend on functions of (...),1(...)k(...)m - are some specially constructed normalized spaces. In addition, in terms of the fulfillment of some non-resonant condition, the paper gives a test of a boundedness of the integral from the product of functions when integrating over a subset of R^n.

Fractional Integral Inequalities for Some Convex Functions

In this paper, we obtained several new integral inequalities using fractional Riemann-Liouville integrals for convex s-Godunova-Levin functions in the second sense and for quasi-convex functions. The results were gained by applying the double Hermite-Hadamard inequality, the classical Holder inequalities, the power mean, and weighted Holder inequalities. In particular, the application of the results for several special computing facilities is given. Some applications to special means for arbitrary real numbers: arithmetic mean, logarithmic mean, and generalized log-mean, are provided.

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].

On Variations of the Neumann Eigenvalues of p-Laplacian Generated by Measure Preserving Quasiconformal Mappings

We study variations of the first nontrivial eigenvalue of the two-dimensional p-Laplace operator, p > 2, generated by measure preserving quasiconformal mappings. The study is based on the geometric theory of composition operators in Sobolev spaces and sharp embedding theorems. Using a sharp version of the reverse Holder inequality, we obtain a lower estimate for the first nontrivial eigenvalue in the case of Ahlfors type domains.

Milne type inequality and interval orders

In this paper, we prove some Milne type inequalities for interval-valued functions and, along with it, we explore some connections with other inequalities. More precisely, using the Aumann integral and the Kulisch–Miranker order and including-order on the space of real and compact intervals, we establish some Milne type inequalities for interval-valued functions. Also, using different orders, we obtain some connections with Chebyshev, Cauchy–Schwarz, and Holder inequality. Finally, some new ideas and results based on submodular measures are explored as well as some examples and applications are presented for illustrating our results.

Chiti-type Reverse Hölder Inequality and Torsional Rigidity Under Integral Ricci Curvature Condition

In this paper, we prove a reverse Holder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of Gamara et al. (Open Math. 13(1), 557–570, 2015) and Colladay et al. (J. Geom. Anal. 28(4), 3906–3927, 2018) from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to p-Laplacian.

Finite-time stability analysis of fractional-order memristive fuzzy cellular neural networks with time delay and leakage term

In this paper, we investigate finite-time stability analysis of fractional-order memristive fuzzy cellular neural networks(MFFCNNs) with time delay and leakage term. MFFCNNs are formulated by virtue of differential inclusion and set-valued map theories. By using generalized Bernoulli inequality and Holder inequality, we derived some new sufficient conditions of finite-time stability for the proposed system. Finally two numerical examples are given to show the effectiveness of the proposed approach.