Log-convex Research Articles

Some properties of the Catalan-Qi function related to the Catalan numbers.

In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan–Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function, and inequalities of the Catalan numbers, the Catalan function, and the Catalan–Qi function. As by-products, an exponential expansion and a double inequality for the ratio of two gamma functions are derived.

Backward uniqueness of stochastic parabolic like equations driven by Gaussian multiplicative noise

One proves here the backward uniqueness of solutions to stochastic semilinear parabolic equations and also for the tamed Navier–Stokes equations driven by linearly multiplicative Gaussian noises. Applications to approximate controllability of nonlinear stochastic parabolic equations with initial controllers are given. The method of proof relies on the logarithmic convexity property known to hold for solutions to linear evolution equations in Hilbert spaces with self-adjoint principal part.

Preservation of Logarithmic Convexity by Positive Operators

We show that the Szasz–Mirakyan and Baskakov operators preserve the logarithmic convexity of functions, whereas the Bernstein and Kantorovich operators do not possess this property.

Log and harmonically log-convex functions related to matrix norms

Log and harmonically log-convex functions related to matrix norms

The Sugeno fuzzy integral of log-convex functions

In this paper, we give an upper bound for the Sugeno fuzzy integral of log-convex functions using the classical Hadamard integral inequality. We present a geometric interpretation and some examples in the framework of the Lebesgue measure to illustrate the results.

Piecewise‐linear approximations for a non‐linear transmission expansion planning problem

This study proposes a new approach to solve the transmission expansion planning (TEP) problem considering that the network is represented by a non-linear model. Non-linear constraints are approximated with piecewise-linear (PWL) functions, which renders TEP a mixed-integer linear programming (MILP) problem. This study presents MILP formulations obtained from the PWL approximation using the incremental, special ordered sets of type 2 (SOS2) and the logarithmic convex combination (Log) model. Three test systems are used to assess the effectiveness of the formulations. The computational analysis encompasses a baseline formulation, which accounts for the costs on line installation and transmission losses, and a variation of this formulation which minimises the operating costs rather than transmission losses. The results obtained by the MILP models are compared with those of a transmission planning model based on the DC representation of the network. The analysis of the results showed that expansion plans obtained from the MILP formulations are feasible, whereas the plans derived from the DC model are infeasible with respect to system constraints.

Logarithmic convexity of the eigenfunction of the Hessian operator in the ball

<p>This paper is concerned with the logarithmic convexity of the eigenfunction of the Hessian operator. It is proved that the logarithm of the eigenfunction of the Hessian operator is strictly convex in the ball.</p>

F-Divergence functional of operator log-convex functions

We investigate the non-commutative -divergence functional of operator log-convex functions. In particular, we show that a continuous function is operator log-convex if and only iffor all continuous fields and of strictly positive operators. Moreover, we define operator -convex functions and investigate some of its properties. In particular, we show that this class contains operator convex functions and operator log-convex functions.

On an interaction function for copulas

We study properties of a local dependence function of Wang for copulas. In this paper this dependence function is called the mixed derivative measure of interactions as it is a mixed derivative of a log of a density function. It is stressed that this measure is not margin free in the sense that the interaction function of a density and the corresponding copula are not equal. We show that there is no Archimedean copula with constant interactions. The interaction function is positive (negative) for an Archimedean copula density whose second derivative of the generator is log convex (log concave). Moreover, the only Archimedean copula with interactions proportional to its density is Frank’s copula. We obtain some preliminary results concerning the connection between the behaviour of the interaction function and the tail dependence of the distribution. Moreover, the notion of an interaction function has been extended to more than two dimensional case, and we study its properties for a canonical Archimedean copula.

On Unique Continuation for Navier-Stokes Equations

We study the unique continuation properties of solutions of the Navier-Stokes equations. We take advantage of rotation transformation of the Navier-Stokes equations to prove the “logarithmic convexity” of certain quantities, which measure the suitable Gaussian decay at infinity to obtain the Gaussian decay weighted estimates, as well as Carleman inequality. As a consequence we obtain sufficient conditions on the behavior of the solution at two different timest0=0andt1=1which guarantee the “global” unique continuation of solutions for the Navier-Stokes equations.

Improvements of the Hermite-Hadamard inequality on time scales

In this paper we give refinements of converse Jensen’s inequality as well as of the Hermite-Hadamard inequality on time scales. We give mean value theorems and investigate logarithmic and exponential convexity of the linear functionals related to the obtained refinements. We also give several examples which illustrate possible applications for our results.

On Hermite-Hadamard type integral inequalities for n-times differentiable log-preinvex functions

In this paper, new Hermite-Hadamard type inequalities for n-times differentiable log-preinvex functions are established. The established results generalize some of those results proved in recent papers for differentiable log-preinvex functions and differentiable log-convex functions.

Cauchy-type means and exponential and logarithmic convexity for superquadratic functions on time scales

n this paper, we define positive functionals by using the Jensen's inequality, converse of Jensen's inequality, and Jensen-Mercer's inequality on time scales for superquadratic functions. We give mean-value theorems and introduce related Cauchy-type means by using the functionals mentioned above and show the monotonicity of these means. We also show that these functionals are exponentially convex and give some applications of them by using the log-convexity and exponential convexity.

Sharp Hardy uncertainty principle and gaussian profiles of covariant Schrödinger evolutions

We prove a sharp version of the Hardy uncertainty principle for Schrödinger equations with external bounded electromagnetic potentials, based on logarithmic convexity properties of Schrödinger evolutions. We provide, in addition, an example of a real electromagnetic potential which produces the existence of solutions with critical gaussian decay, at two distinct times.

Improvements of Jensen-Type Inequalities for Diamond-α Integrals.

We give further improvements of the Jensen inequality and its converse on time scales, allowing also negative weights. These results generalize the Jensen inequality and its converse for both discrete and continuous cases. Further, we investigate the exponential and logarithmic convexity of the differences between the left-hand side and the right-hand side of these inequalities and present several families of functions for which these results can be applied.

Lower bounds of end effects for a nonhomogeneous isotropic linear elastic solid in anti-plane shear

In this paper we give lower bounds for the spatial decay of the solutions for anti-plane shear deformations in the case of isotropic inhomogeneous elastic materials. We first consider the case when the shear modulus only depends on the lateral direction. By means of the logarithmic convexity arguments we obtain the required estimates. Some pictures illustrate our results. We also study the general inhomogeneity. We give some lower bounds whenever shear modulus satisfies several requirements.

STOLARSKY-TYPE MEANS RELATED TO GENERALIZATIONS OF STEFFENSEN’S AND GAUSS’ INEQUALITY

In this paper, functionals defined as the difference between the left-hand and the right-hand sides of generalizations of Steffensen’s and Gauss’ inequality are studied. n-exponential and exponential as well as logarithmic convex functions related to defined functionals are produced. Furthermore, several families of exponentially convex functions are investigated, and new Stolarsky-type means related to defined functionals are generated.

More accurate weak majorization relations for the Jensen and some related inequalities

Motivated by results of Aujla and Silva [3], we give several more precise weak majorization and eigenvalue inequalities for some matrix versions of the famous Jensen inequality with regard to a convexity. Our main results are then applied to log convex functions. As an application, we obtain refinements of some well-known matrix inequalities.

An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind

In the paper, the author establishes an integral representation, finds the complete monotonicity, minimality, and logarithmic convexity, and presents some inequalities of Cauchy numbers of the second kind.

Some fixed point theorems and common fixed point theorem in log-convex structure

Some fixed point theorems and common fixed point theorem in Logarithmic convex structure areproved.