Log-convex Research Articles

Qualitative properties for a Moore–Gibson–Thompson thermoelastic problem with heat radiation

In this work, we study some qualitative properties arising in the solution of a thermoelastic problem with heat radiation. The so-called Moore–Gibson–Thompson equation is used to model the heat conduction. By using the logarithmic convexity argument, the uniqueness and instability of solutions are proved without imposing any condition on the elasticity tensor. Then, the existence of solutions is obtained assuming that the elastic tensor is positive definite applying the theory of linear semigroups, and the exponential energy decay is shown in the one-dimensional case. Finally, we consider the one-dimensional quasi-static version and we assume that the elastic coefficient is negative. The existence and decay of solutions are proved, and a justification of the quasi-static approach is also provided.

Convexity of multiplicities of filtrations on local rings

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

Isoperimetric inequalities and regularity of A-harmonic functions on surfaces

We investigate the logarithmic and power-type convexity of the length of the level curves for a-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the p-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} regularity. The results are complemented by a number of estimates for the derivatives L′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L'$$\\end{document} and L′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L''$$\\end{document} of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes results due to Alessandrini, Longinetti, Talenti and Lewis in the Euclidean setting, as well as a recent article of ours devoted to the harmonic case on surfaces.

Multiple-term improvements of Jensen's inequality for (p,h)-convex and (p,h)-log convex functions

Multiple-term improvements of Jensen's inequality for (p,h)-convex and (p,h)-log convex functions

MATRIX FEJÉR AND LEVIN-STEČKIN INEQUALITIES

Fejér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For the matrix Levin-Stečkin type, we present more rigorous results involving the partial Löewner ordering for Hermitian matrices. Further related results involving synchronous functions are presented, too.

Inverse problems for general parabolic systems and application to Ornstein-Uhlenbeck equation

We investigate the link between inverse problems and final state observability for a general class of parabolic systems. We generalize a stability result for initial data known for the case of self-adjoint dissipative operators. More precisely, we consider a system governed by an analytic semigroup. Under the assumption of final state observability, we prove a logarithmic stability estimate depending on the analyticity angle of the semigroup. This is done by using a general logarithmic convexity result. The abstract result is illustrated by considering the Ornstein–Uhlenbeck equation.

On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers

In the paper, by virtue of Wronski's formula and Kaluza's theorem related to a power series and its reciprocal, by means of Cahill and Narayan's recursive relation, and with the aid of the logarithmic convexity of the sequence of the Bernoulli numbers, the author presents the signs of certain Toeplitz-Hessenberg determinants whose elements involve the Bernoulli numbers and combinatorial numbers. Moreover, with the help of a derivative formula for the ratio of two differentiable functions, the author provides an alternative proof of Wronski's formula.

The backward problem for time-fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations ∂ t α u ( t ) = Au ( t ) with the Caputo derivative of order 0 < α ≤ 1 , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.

Impulse controllability for degenerate singular parabolic equations via logarithmic convexity method

Abstract In this paper, we study null approximate controllability of degenerate singular parabolic equations under the action of an impulsive control. To this aim, we prove an observation estimate at one point in time for the problems associated to the operators: $$ \begin{align*}&amp; u_{t} -(x^{\alpha} u_{x})_{x} - \dfrac{\mu}{x^{\beta}} u = 0, \qquad x \in \left(0, 1\right), \end{align*} $$ where the parameters $\alpha \geq 0$, $\beta , \mu \in \mathbb{R}$ satisfy suitable assumptions. The method of proof combines both the logarithmic convexity and the Carleman commutator.

Increasing property and logarithmic convexity of functions involving Dirichlet lambda function

Abstract In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.

Some inequalities bounding certain ratios of the q-gamma function

In this short paper, we established q- analogues of some well-known inequalities for the classical Gamma function and some inequalities bounding the ratio Γ_q (x)/Γ_q (y), where Γ_q (.) is the q-analogue of the Gamma function. The method employed in presenting the results made use of strict logarithmic convexity of a function and classical mean value theorem involving the q-analogue of the psi function. At the end, the proven result provided the q-analogue of some known result concerning the classical Gamma function.

Finite-Time Stabilization and Impulse Control of Heat Equation with Dynamic Boundary Conditions

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms.

Stabilization estimates for the Brinkman–Forchheimer–Kelvin–Voigt equation backward in time

The final value value problem for the Brinkman–Forchheimer–Kelvin–Voigt equations is analysed for quadratic and cubic types of Forchheimer nonlinearity. The main term in the Forchheimer equations is allowed to be fully anisotropic. It is shown that the solution depends continuously on the final data provided the solution satisfies an a priori bound in L^3. The technique employed avoids the use of a specialist method for an improperly posed problem such as logarithmic convexity.

Impulse controllability for the heat equation with inverse square potential and dynamic boundary conditions

Abstract In this paper, we investigate the null approximate impulse controllability of the heat equation with an inverse square potential subject to dynamic boundary conditions in the ball $B(0, R_{0})$ of radius $R_{0}=\left (\frac{4}{3}\right )^{\frac{3}{2}}$. To that purpose, we use the Carleman commutator approach to show a logarithmic convexity estimate traducing an observability inequality at one instant of time.

Study of Log Convex Mappings in Fuzzy Aunnam Calculus via Fuzzy Inclusion Relation over Fuzzy-Number Space

In this paper, with the use of newly defined class up and down log–convex fuzzy-number valued mappings, we offer a few new and original mappings defined by applying some mild restrictions over the definition of up and down log–convex fuzzy-number valued mapping. With the use of these mappings, we are able to develop partners of Fejér-type inequalities for up and down log–convexity, which improve upon certain previously established findings. The discussion also includes these mappings’ characteristics. Moreover, some nontrivial examples are also provided to prove the validation of our main results.

Log concavity preservation by beta operator based on probability tools

A number of papers have dealt with the preservation of log convexity and log concavity based on various operators. For instance, in Badía and Sangüesa [10], the preservation of log convexity and log concavity under Bernstein operators was discussed based on some characteristics of a stochastic process. However, regarding beta-type operators, the preservation of log concavity does not hold based on such a probabilistic method used in Badía and Sangüesa [10]. In this study, we explore the preservation of log concavity for the beta operator using alternative probabilistic tools. Notably, we show results on the preservation of log concavity for monotone log concave functions. Further, some results of application to some specific functions, ageing classes of the deterioration Dirichlet process, related operators and order statistics are provided.

Increasing property and logarithmic convexity concerning Dirichlet beta function, Euler numbers, and their ratios

In the paper, by virtue of an integral representation of the Dirichlet beta function, with the aid of a relation between the Dirichlet beta function and the Euler numbers, and by means of a monotonicity rule for the ratio of two definite integrals with a parameter, the author finds increasing property and logarithmic convexity of two functions and two sequences involving the Dirichlet beta function, the Euler numbers, and their ratios.

Optimization of Buffer Networks via DC Programming

This brief is concerned with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H^{2}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H^{\infty }$ </tex-math></inline-formula> norm-constrained optimization problems of dynamic buffer networks. The extended network model is introduced first, wherein the weights of all edges can be tuned independently. Because of the emerging nonconvexity of the extended model, previous results of positive linear systems failed to address this situation. By resorting to the log–log convexity of a class of nonlinear functions called posynomials, the optimization problems can be reduced to differential convex programming problems. The proposed framework is illustrated for large-scale networks.

New inequalities for (p,h)-convex functions for τ-measurable operators

The main goal of this article is to present new inequalities for (p, h)-convex and (p,h) log-convex functions for a non-negative super-multiplicative and super-additive function h. Our first main result will be h? (v/?) ? (h(1 ? v) f (a) + h(v) f (b))? ? f ? [((1 ? v)ap + vbp) 1 p] / (h(1 ? ?) f (a) + h(?) f (b))? ? f ? [((1 ? ?)ap + ?bp)1p] ? h? (1 ? v/1 ? ?), for the positive (p, h)-convex function f, when ? ? 1, p ? R\{0} and 0 ? v ? ? ? 1. This gives a generalization of an important result due to M. Sababheh [Linear Algebra Appl. 506 (2016), 588-602]. As applications of our results, we present many inequalities for the trace, and the symmetric norms for ?-measurable operators.

Power inequalities for log-convex functions with applications

In this paper, we present new inequalities for log-convex functions, with some applications to operator means. The significance of the obtained results is two folded; the results themselves and the way they extend many known results in the literature.