Inequalities For Functions Research Articles

Hypoelliptic functional inequalities

In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Sobolev, Rellich, Hardy–Littllewood–Sobolev, Gagliardo–Nirenberg, Caffarelli–Kohn–Nirenberg and Heisenberg–Pauli–Weyl type uncertainty inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous Lie groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators.

Report of Meeting: The Twenty-third Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 – February 3, 2024

Report of Meeting: The Twenty-third Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 – February 3, 2024

Exploring nine different proofs of a famous logarithmic inequality

This article explores and analyzes nine different proofs of a fundamental logarithmic inequality, say "log(1+x)/x ≥ 1/ (1+x/2) for x>-1". Some proofs are already published; others are new or consider new approaches or new angles of research. They are based on various techniques, such as differentiation, series expansion, and the application of well-known inequalities such as the primitive, Cauchy-Schwarz, logarithmic mean, hyperbolic tangent function, Jensen, and Hermite-Hadamard inequalities. A graphical work illustrates some results. Therefore, our study clarifies the different mathematical foundations of this seemingly straightforward but important inequality and goes beyond it with some new material.

The Laguerre inequality and determinantal inequality for the broken k-diamond partition function

The Laguerre inequality and determinantal inequality for the broken k-diamond partition function

A study on error bounds for Newton-type inequalities in conformable fractional integrals

Abstract The authors of the paper suggest a novel approach in order to examine an integral equality using conformable fractional operators. By using this identity, some Newton-type inequalities are proved for differentiable convex functions by taking the modulus of the newly established equality. Moreover, we prove some Newton-type inequalities by using the Hölder and power-mean inequality. Furthermore, some new results are presented by using special choices of obtained inequalities. Finally, we give some conformable fractional Newton-type inequalities for functions of bounded variation.

Class of bounds of the generalized Volterra functions

Abstract In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

Complex Hanner’s inequality for many functions

Complex Hanner’s inequality for many functions

Exponential stability of non-conformable fractional-order systems

Abstract Recently, the authors Guzman et al. (2018) introduced a new simple well-behaved definition of the fractional derivative called non-conformable fractional derivative. In this paper we study the exponential stability of non-conformable fractional-order systems by using the Lyapunov function and Gronwall inequality. These inequalities can be used as handy tools to research stability problems of nonlinear systems. Sufficient conditions for exponential stability are given using the Lyapunov theory. Further, deals with the state feedback stabilization problems for a family of nonlinear systems satisfying a Lipschitz continuity condition. The stability of the controller is proved by means of the new Lyapunov stability theorem given in this paper. A numerical example is given to illustrate the efficiency of the obtained result.

An inequality for log-concave functions and its use in the study of failure rates

Abstract We establish here an integral inequality for real log-concave functions, which can be viewed as an average monotone likelihood property. This inequality is then applied to examine the monotonicity of failure rates.

Norm inequalities for functions of matrices

In this paper, we prove several spectral norm and unitarily invariant norm inequalities for matrices in which the special cases of our results present some known inequalities. Also, some of our results give interpolating inequalities which are related to well-known recent inequalities.

Quantum simpson like type inequalities for q-differentiable convex functions

Quantum simpson like type inequalities for q-differentiable convex functions

Event-triggered H∞ control for uncertain discrete-time singular systems via sliding mode approach

This article analyzes the event-triggered (ET) scheme for uncertain discrete-time singular systems via [Formula: see text] sliding mode control (SMC). The issue of network delay is taken into account, and a Bernoulli stochastic variable is used to explain its distribution feature. A sliding mode surface (SMS) function is built by using ET scheme, and the corresponding sliding mode dynamics (SMD) is combined. Then, based on Lyapunov-Kravoskii functional (LKF) stability theory and linear matrix inequality (LMI) technology, [Formula: see text]-admissibility criterion for closed-loop SMD is deduced. Additionally, an improved SMC rule based on ET strategy is created so that the system trajectories can be driven onto the prescribed SMS in finite time. Finally, the effectiveness of the approach is demonstrated by three simulated examples.

Sharp Hardy-Leray inequality for solenoidal fields

We compute the best value of the constant in functional integral inequality called the Hardy-Leray inequality for solenoidal vector fields on RN. This gives a solenoidal improvement of the original best constant known for unconstrained vector fields, and refines the former work by Costin-Maz'ya [4] who found a new best constant in Hardy-Leray inequality for axisymmetric solenoidal vector fields. Our method does not require any symmetry assumption to derive the same constant number as Costin-Maz'ya's. Moreover, the constant has a simpler expression in relation to the weight exponent and turns out to be unattainable in the space of functions satisfying some regularity condition fitting to the Hardy-Leray inequality.

Sharp coefficient inequalities of starlike functions connected with secant hyperbolic function

This article comprises the study of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} that represents the class of normalized analytic functions f satisfying ςf′(z)/f(ς)≺sech(ς)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\varsigma \\mathsf{f}}^{\\prime }(z)/\\mathsf{f}( {\\varsigma })\\prec \\sec h ( \\varsigma ) $\\end{document}. The geometry of functions of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document}. We also investigate the same sharp results for inverse coefficients.

On the gradient rearrangement of functions

AbstractIn this paper, we introduce a symmetrization technique for the gradient of a $$\text {BV}$$ BV function, which separates its absolutely continuous part from its singular part (sum of jump and Cantorian part). In particular, we prove a $$\text {L }^{\text {1}}$$ L 1 comparison between the function and the symmetrization just mentioned. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

The Stability of Multi-Coefficients Pexider Additive Functional Inequalities in Banach Spaces

The Hyers–Ulam stability of multi-coefficients Pexider additive functional inequalities in Banach spaces is investigated. In order to do this, the fixed point method and the direct method are used.

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

A recurrent neural network approach for nonconvex interval quadratic programming

In the paper, a new approach is presented to tackle an interval optimization problem with nonconvex interval objective function and affine interval equality and inequality constraints utilizing a recurrent neural network. The approach involves converting the nonconvex interval quadratic programming into a cubic optimization via convex combinations and considering its weight as a decision variable. The recurrent neural network is then established to solve the cubic optimization through subgradient techniques and an exterior penalty function method. Moreover, it is proven that there is a unique global solution to the recurrent neural network, and its trajectory can reach the feasible region in a finite time and remain there. Additionally, it is demonstrated that the trajectory of recurrent neural network converges exponentially or in finite time towards a singleton belonging to the set of critical points of the cubic optimization. This approach is beneficial for interval quadratic programming problems where the weight is unknown or challenging to determine in practical applications. Finally, the effectiveness of this approach is verified through two numerical examples.

Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$-range

Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with $${\varepsilon }$$-range

Hecke-type series involving infinite products

In this paper, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain truncated versions of these series as well as truncated forms of some other known series of such types. Finally, as an application, we deduce six infinite families of inequalities for various partition functions. Our proofs of the main results heavily rely on a formula from the work of Liu (2013).