Holder Inequality Research Articles

A Refinement and a divided difference reverse of Jensen's inequality with applications

A refinement and a new sharp reverse of Jensen's inequality for convex functions in terms of divided diferences is obtained. Applications for means, the Holder inequality and for f-divergence measures in information theory are also provided.

The sharpening Hölder inequality via abstract convexity

In this work, a new inequality by sharpening the well-known Holder inequality by means of a theorem based on abstract convexity is derived.

Existence of solutions for fractional differential equations with nonlocal and average type integral boundary conditions

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for a boundary value problem of fractional differential equations with nonlocal and average type integral boundary conditions. The Leray–Schauder nonlinear alternative, Krasnoselskii’s fixed point theorem and Banach’s fixed point theorem together with Holder inequality are applied to construct proofs for the main results. Examples illustrating the obtained results are also presented.

High–low frequency slaving and regularity issues in the 3D Navier–Stokes equations

The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the $3D$ Navier-Stokes equations. A set of dimensionless frequencies $\{\tilde{\Omega}_{m}(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the $\tilde{\Omega}_{m}$ ($m>1$) are slaved to $\tilde{\Omega}_{1}$ (the global enstrophy) in the form $\tilde{\Omega}_{m} = \tilde{\Omega}_{1}\mathcal{F}_{m}(\tilde{\Omega}_{1})$. This is shaped by the constraint of two Holder inequalities and a time average from which emerges a form for $\mathcal{F}_{m}$ which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions $1 \leq \lambda_{m}(\tau) \leq 4$, where curves of constant $\lambda_{m}$ form the boundaries between tongue-shaped regions. In regions where $2.5 \leq \lambda_{m} \leq 4$ and $1 \leq \lambda_{m} \leq 2$ the Navier-Stokes equations are shown to be regular\,: numerical simulations appear to lie in the latter region. Only in the central region $2 < \lambda_{m} < 2.5$ has no proof of regularity been found.

Prove inequalities by solving maximum/minimum problems using a computer algebra system

This presentation offers an alternative to traditional approaches to proving non-trivial inequalities, such as applying AM-GM, Cauchy, Holder and Minkowski inequalities. This alternative approach, as demonstrated by various examples, establishes the validity of an inequality through solving a maximization/minimization problem by commonly practiced procedures in Calculus. Since the procedures are algorithmic, a Computer Algebra System (CAS) can carry out the computation efficiently.

Sobolev regularity for the first order Hamilton–Jacobi equation

We provide Sobolev estimates for solutions of first order Hamilton–Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse Holder inequality. Applications to mean field games are discussed.

Holder's Inequalities for a New Class of P-Valent Analytic Functions

In this paper, we introduced a new class of p-valent analytic functions using a linear multiplier Dziok-Srivastava operator D m;q;s f(z)(m2 N0 =f0; 1;:::g;

More accurate Young, Heinz, and Hölder inequalities for matrices

In this paper we deal with a more precise estimates for the matrix versions of Young, Heinz, and Holder inequalities. First we give an improvement of the matrix Heinz inequality for the case of the Hilbert–Schmidt norm. Then, we refine matrix Young-type inequalities for the case of Hilbert–Schmidt norm, which hold under certain assumptions on positive semidefinite matrices appearing therein. Finally, we give the refinement and the reverse of the matrix Holder inequality which holds for every unitarily invariant norm. As applications, we also obtain improvements of some well-known matrix inequalities in a quotient form. Our results are compared with the previously known from the literature.

On Jensen’s inequality, Hölder’s inequality, and Minkowski’s inequality for dynamically consistent nonlinear evaluations

In this paper, the dynamically consistent nonlinear evaluations that were introduced by Peng are considered in probability space $L^{2} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{t\geq0},P )$ . We investigate the n-dimensional ( $n\geq1$ ) Jensen inequality, Holder inequality, and Minkowski inequality for dynamically consistent nonlinear evaluations in $L^{1} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{t\geq0},P )$ . Furthermore, we give four equivalent conditions on the n-dimensional Jensen inequality for g-evaluations induced by backward stochastic differential equations with non-uniform Lipschitz coefficients in $L^{p} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{0\leq t\leq T},P )$ ( $1< p\leq2$ ). Finally, we give a sufficient condition on g that satisfies the non-uniform Lipschitz condition under which Holder’s inequality and Minkowski’s inequality for the corresponding g-evaluation hold true. These results include and extend some existing results.

Nonlocal self-improving properties

Solutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by integral(Rn)integral(Rn)[u(x) - u(y)][eta(x) - eta(y)]K(x, y) dx dy = integral(Rn) f eta dx for all eta is an element of C-c(infinity) (R-n), where the kernel K( . ) is a measurable function and satisfies the bounds 1/Lambda vertical bar x - y vertical bar(n+2 alpha) <= K(x, y) <= Lambda/vertical bar x - y vertical bar(n+2 alpha) with 0 1, while f is an element of L-loc(q)(R-n) for some q > 2n/(n + 2 alpha). The main result states that there exists a positive, universal exponent delta equivalent to delta(n, alpha, Lambda, q) such that for every weak solution u the self-improving property u is an element of W-alpha,W-2 (R-n) double right arrow u is an element of W-loc(alpha+delta,2+delta) (R-n) holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Holder-type inequalities in R-2n and which is implied by delicate covering and exit-time arguments. In turn, such reverse Holder inequalities are based on the concept of dual pairs, that is, pairs (mu, U) of measures and functions in R-2n which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order alpha.

Strengthenings of the Lyapunov, Hölder, and Minkowski Inequalities

Some strengthenings and generalizations of the classical Lyapunov, Holder, and Minkowski inequalities are given in probabilistic and also in analytical terms. Bibliography: 11 titles.

Oscillation of higher order functional differential equations with positive and negative coefficients and damping term

The oscillation for a class of even order nonlinear variable delay damped functional differential equation with positive and negative coefficients and nonlinear neutral term was discussed.By introducing parameter function and the generalized Riccati transformation,using Holder inequality and some necessary analytic techniques,some new criteria for the oscillation of the equation were proposed.These criteria improve and generalize some known results.Examples were given to illustrate the main results.

Sharp weighted bounds for one-sided maximal operators

In this note we prove some results about the best constants for the boundedness of the one-sided Hardy–Littlewood maximal operator in \(L^p(\mu )\), where \(\mu \) is a locally finite Borel measure, that in the two-sided weights have been obtained by Buckley (Trans Am Math Soc 340(1):253–272, 1993) and more recently by Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012). To prove Bucley’s theorem for one-sided maximal operators, we follow the ideas of Lerner (Proc Am Math Soc 136(8):2829–2833, 2008). To obtain a better estimate in terms of mixed constants we follow the steps in Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012) i.e., (a) getting a sharp estimate for the constant for the weak type type, in terms of the one-sided \(A_p\) constant, (b) obtaining a sharp reverse Holder inequality and (c) using Marcinkiewicz interpolation theorem. Our proofs of these facts are different from those in Hytonen and Perez (Anal PDE 6(4):777–818, 2013) and Hytonen et al. (J Funct Anal 263(12):3883–3899, 2012) and apply to more general measures.

Several refinements and counterparts of Radon's inequality

We establish that the inequality of Radon is a particular case of Jensen's inequality. Starting from several refinements and counterparts of Jensen's inequality by Dragomir and Ionescu, we obtain a counterpart of Radon's inequality. In this way, using a result of Simic we find another counterpart of Radon's inequality. We obtain several applications using Mortici's inequality to improve Holder's inequality and Liapunov's inequality. To determine the best bounds for some inequalities, we used Matlab program for different cases.

Converses of Copson's inequalities on time scales

In this paper, we will prove some new dynamic inequalities on a time scale T . These inequalities when T = N contain the discrete inequalities due to Bennett and Leindler which are converses of Copson’s inequalities. The main results will be proved using the Holder inequality and Keller’s chain rule on time scales. Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40. 34N05.

A modification of the method of fictitious domains for stationary model of non-Newtonian liquids

The substantiation of one modification method of fictitious domains with continuation on junior coefficient for the stationary equations of non-Newtonian liquids is given in this work. The domain is three-dimensional and bounded. A theorem of an existence and convergence of the generalized solution of an auxiliary problem is proved. The theorem is proved by the method of a priori estimates. Inequality of imbedding theorems, Holder inequality and Young's inequality are used. The sequence of approximate solutions is constructed using the Galerkin method. The limit in the integral identity through the selected sequence. For solutions obtained uniform evaluation standards of functional spaces. An estimation of convergence of the strong solution of the approximating problem is deduced.

Some General Inequalities for Choquet Integral

With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f1, f2, …, fn are comonotone, we also obtain the Holder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.

A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

A quantitative two weight theorem for the Hardy-Little- wood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in (HP) and (HPR12) is obtained which avoids the use of the sharp quantitative reverse Holder inequality for A1 proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the non- empty annuli condition.

The existence of symmetric positive solutions for a seconder-order difference equation with sum form boundary conditions

Abstract In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem Δ ( g ( k − 1 ) Δ u ( k − 1 ) ) + w ( k ) f ( k , u ( k ) ) = 0 subject to the boundary conditions: a u ( 0 ) − b g ( 0 ) Δ u ( 0 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , a u ( n ) + b g ( n − 1 ) Δ u ( n − 1 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , where a , b &gt; 0 , Δ u ( k ) = u ( k + 1 ) − u ( k ) for k ∈ { 0 , 1 , … , n − 1 } , g ( k ) &gt; 0 is symmetric on { 0 , 1 , … , n − 1 } , w ( k ) is symmetric on { 0 , 1 , … , n } , f : { 0 , 1 , … , n } × [ 0 , + ∞ ) is continuous, f ( k , u ) = f ( n − k , u ) for all ( k , u ) ∈ { 0 , 1 , … , n } × [ 0 , + ∞ ) , and h ( i ) is nonnegative and symmetric on { 0 , 1 , … , n } . By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.