Inequalities For Functions Research Articles

Tensorial and Hadamard product inequalities for functions of selfadjoint operators in Hilbert spaces in terms of Kantorovich ratio

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with 0 <γ ≤ f(t)/g(t)≤ Γ for t ∈ I and if A and B are selfadjoint operators with Sp (A), Sp (B) ⊂ I, then [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)] ≤ (1 − ν) f (A) ⊗ g (B) + ν g (A) ⊗ f (B) ≤ [(γ + Γ) 2/4γΓ]R [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)]. The above inequalities also hold for the Hadamard product “ ◦ ” instead of tensorial product “ ⊗ ”.

Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

We consider conservative and gradient flows for N -particle Riesz energies with mean-field scaling on the torus \mathbb{T}^d , for d\geq 1 , and with thermal noise of McKean–Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. (2019, 2019–2020, 2020) and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range d-2\leq s<d not addressed by the previous work Rosenzweig and Serfaty (2023). We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang (2018) for so-called \dot{W}^{-1,\infty} kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al. (2021).

Fractional dual Simpson-type inequalities for differentiable s-convex functions

Abstract In this paper, a new integral identity is provided. Based on this equality, Simpson-type dual integral inequalities for functions whose first derivatives are s-convex via Riemann–Liouville fractional integrals are established.

New Versions of Fuzzy-Valued Integral Inclusion over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman’s Integral Inequalities

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings (FNVMs) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of p-convexity over up and down (UD) fuzzy relation has been introduced which is known as UD-p-convex fuzzy number-valued mappings (UD-p-convex FNVMs). We offer a thorough analysis of Hermite–Hadamard-type inequalities for FNVMs that are UD-p-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for p-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples.

Generalization of Hermite-Hadamard Inequality for Twice Differentiable Convex and Quasi-Convex Functions

Generalization of Hermite-Hadamard Inequality for Twice Differentiable Convex and Quasi-Convex Functions

Inference in models with partially identified control functions

In multiple contributions to the literature, James L. Powell and coauthors have developed estimators for semiparametric models where sample selectivity and/or endogeneity can be handled through a “control function”. Their methods rely on pairwise comparisons of observations which match (asymptotically) the control functions. Conditional on this matching, a moment condition can identify the parameters of the model. However, there exist instances where the control functions are unobserved, but we have bounds for them which depend on observable covariates. These bounds can arise directly from the nature of the data available (e.g, with interval data), or they can be derived from an economic model. The inability to observe the control functions precludes the matching proposed in Powell’s methods. In this paper we show that, under certain conditions, testable implications can still be obtained through pairwise comparisons of observations for which the control-function bounds are disjoint. Testable implications now take the form of pairwise functional inequalities. We propose an inferential procedure based on these pairwise inequalities and we analyze its properties.

Ostrowski-Type Fractional Integral Inequalities: A Survey

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (β,m)-convex functions, exponential-convex functions, ζ¯,β,γ,δ-convex functions, quasi-geometrically convex functions, s−e-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented.

Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.

Subadditive and Superadditive Inequalities for Convex and Superquadratic Functions

Convex functions and their analogues have been powerful tools in almost all mathematical fields, including optimization, fractional calculus, mathematical analysis, functional analysis, operator theory, and mathematical physics. It is well established in the literature that a convex function $f:[0,\infty)\to [0,\infty)$ with $f(0)=0$ is necessarily superadditive, while a concave function $f:[0,\infty)\to [0,\infty)$ is subadditive. The converses of these two assertions are not valid in general. The main target of this article is to study the subadditivity and superadditivity of convex and superquadratic functions. In particular, we obtain several results extending, refining, and reversing some known inequalities in this direction. Further discussion of superquadratic functions in this line will be given.

Two New Functional Inequalities and Their Application to the Eventual Smoothness of Solutions to a Chemotaxis-Navier–Stokes System with Rotational Flux

Two New Functional Inequalities and Their Application to the Eventual Smoothness of Solutions to a Chemotaxis-Navier–Stokes System with Rotational Flux

Tradeoff Techniques for Stabilization of Network-Controlled DC Motor Speed Control System with Uncertainties

This paper details the development of a tradeoff techniques using the conventional time-domain method for stability assurance of network-controlled DC motor speed control system with uncertainties. The DC motor controlled using an open communication network introduces time-delays in the control loop. These time-delays are one of the factors that decides the stability and performance of the dynamical systems. Stabilization of network-controlled DC motor is achieved by tuning the network controller with concern to time-delay. The maximum extend of time-delay called delay margin is computed by developing stability criterion using Lyapunov–Krasvoskii functional and linear matrix inequality techniques. The impact of load disturbances on stability delay margins are investigated and further determining the relationship between delay margin and controller gains for network-controlled DC motor gives a tradeoff between stability and dynamic performance. Finally, proposed results for network-controlled DC motors are validated through Matlab simulation and experimental setup.

Plant functional traits and tree size inequality improved individual tree height prediction of mid-montane humid evergreen broad-leaved forests in southwest China

Tree height and diameter at breast height (DBH) are key survey factors in forest inventory. Compared to measuring DBH, the process of measuring tree height is time-consuming, labor-intensive and costly, especially in natural mixed forests. Tree height is therefore commonly predicted using height-diameter (H-D) models. Nevertheless, developing H-D models for natural mixed forests are hampered by the large diversity of tree species and the small number of rare tree species observed. In this study, a nonlinear mixed effects H-D model was established for a mid-montane humid evergreen broad-leaved forest in southwest China. For this purpose, we classified tree species by K-means clustering according to the specific plant functional traits. Data were collected from 5, 737 trees in 100 subplots with 86 species. The results revealed that the nonlinear mixed effects model of tree height accounted for 75% of the variation in tree height without significant heteroscedasticity, and that the model parameters had biological significance. In the 10-fold cross validation, no clear fluctuation in the model evaluation metrics was observed, indicating the absence of overfitting in the model. Of the 15 stand variables, dominant height (Hd) and Margalef’s index of DBH (DMg), which contributed the most to the height-diameter model, were selected to expand the model. Our findings suggest that tree height increased with increasing values of Hd and DMg, and vice versa. This study highlights the feasibility of categorizing natural mixed forests by utilizing plant functional traits and the influential role of tree size inequality on tree height variation.

Trudinger‐type inequalities for variable Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over metric measure spaces

AbstractWe study Trudinger‐type inequalities for variable Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over bounded metric measure spaces. As a good example, we obtain Trudinger‐type inequalities for double‐phase functionals . As an application, we introduce Musielak–Orlicz–Morrey–Hajłasz–Sobolev spaces and give Trudinger‐type inequalities for Sobolev functions.

New family of bi-univalent functions with respect to symmetric conjugate points associated with Borel distribution

Abstract In this paper, we introduce and investigate a new family, denoted by 𝒲Σ sc (λ, η, δ, r), of normalized holomorphic and bi-univalent functions with respect to symmetric conjugate points, defined in 𝕌, by making use the Borel distribution series, which is associated with the Horadam polynomials. We derive estimates on the initial Taylor-Maclaurin coefficients and solve the Fekete-Szeg˝o type inequalities for functions in this family.

Fault estimation for nonlinear parameter-varying time-delayed systems

A fault estimation method for a class of nonlinear parameter-varying systems subject to time-varying delay and unmeasured nonlinearities is presented. The unmeasured time-varying parameters are effectively handled using a sector-based condition approach. A gain-scheduling intermediate estimator is proposed to simultaneously estimate the system state and the unknown faults. Design conditions are derived based on Lyapunov–Krasovskii functional and integral inequality techniques. These conditions, expressed as linear matrix inequalities, ensure that the estimation error dynamics are input-to-state stable with respect to the time-derivative of the faults. Moreover, it is demonstrated that for the case of piecewise constant faults, the estimation error dynamics are exponentially stable. As a corollary result, conditions are also presented to design gain-scheduling intermediate estimators for nonlinear parameter-varying systems without time-varying delays. Three physically motivated examples are provided to demonstrate the effectiveness and practical interests of the proposed nonlinear estimation method.

On Coefficient Inequalities of Starlike Functions Related to the q-Analog of Cosine Functions Defined by the Fractional q-Differential Operator

This article extends the study of q-versions of analytic functions by introducing and studying the association of starlike functions with trigonometric cosine functions, both defined in their q-versions. Certain coefficient inequalities like coefficient bounds, Zalcman inequalities, and both Hankel and Toeplitz determinants for the new version of starlike functions are investigated. It is worth mentioning that most of the determined inequalities are sharp with the support of relevant extremal functions.

On shifting the principal eigenvalue of Dirichlet problem to infinity with non-transversal incompressible drift

We prove that it is always possible to add some divergence free drift vector field to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector fields vanish on the boundary and their flow lines individually stay away from the boundary. The capacity of those drift vector fields to accelerate diffusivity originates from high frequency oscillation of the associated flow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for flow invariant functions.

Socioeconomic inequalities in health-related functioning among people with Type 2 Diabetes

Abstract Background Type 2 Diabetes mellitus (T2DM) is a common chronic disease which disproportionally affects disadvantaged groups. People with a low socioeconomic position (SEP) have increased risk of T2DM and people with a low SEP and T2DM have higher HbA1c levels and a higher risk to develop T2DM complications compared to their high SEP counterparts. The aim of this study is to analyze longitudinal socioeconomic differences in health-related functioning in people living with T2DM. Methods Cross-sectional and longitudinal data from 2,004 participants of The Maastricht Study with T2DM were used. SEP was determined by baseline measures of education, occupation and income. Health-related functioning (physical, mental and social) was measured with the SF-36 and the Impact on Participation and Autonomy survey (scores 0-100). Associations of SEP and health-related functioning were studied over a 10-year period (baseline 2010-2018) using linear mixed methods adjusting for demographics, disease characteristics and lifestyle factors. Results Low SEP participants had significantly worse health-related functioning compared to those with a high SEP. For example, participants with low income had lower scores for physical (-4.00[CI 2.62;-5.38]), mental (-2.81[-1.52;-4.10]) and social functioning (-9.10[-6.35;-11.85]) compared to high-income participants. In addition, participants with a low education significantly declined more in mental (interaction estimate time -0.21[-0.05;-0.37]) and social functioning (-0.39[-0.02;-0.76]) compared to participants with high education. Conclusions In individuals with T2DM, socioeconomic health differences are apparent in health-related functioning. Moreover, differences in mental and social functioning widened even more between different levels of education. Beyond disease characteristics and lifestyle, more attention is needed for socioeconomic differences in health-related functioning for people living with T2DM, both in research and practice. Key messages • Among people with T2DM, those with a lower SEP have worse health-related function then individuals with a higher SEP, even with similar demographics, disease characteristics and lifestyle patterns. • Additionally, people with T2DM and low education decline more over time in mental and social functioning than people with T2DM and high education.

The Caffarelli–Kohn–Nirenberg inequalities for radial functions

The Caffarelli–Kohn–Nirenberg inequalities for radial functions

A quantitative stability result for the Prékopa–Leindler inequality for arbitrary measurable functions

We prove that if a triplet of functions satisfies almost-equality in the Prékopa–Leindler inequality, then these functions are close to a common log-concave function, up to multiplication and rescaling. Our result holds for general measurable functions in all dimensions, and provides a quantitative stability estimate with computable constants.