Abstract. A relevant application of the stochastic convex order is the well-known weighted Hermite-Hadamard inequality, where the weight is provided by a given probability distribution. Our goal is to show that, starting from such a fixed weight, we can fill the whole space between the Hermite-Hadamard bounds by highlighting some parametric families of probability distributions. Thus, we propose two alternative constructions based on the convex ordering properties.Mathematics subject classification (2010): 26A51, 26B25, 26D10.

The Hamilton-Poisson geometry has proved to be an interesting approach for a lot of dynamics arising from different areas like biology (Gümral and Nutku, 1993), economics (Dănăiasă et al., 2008), or engineering (Ginoux and Rossetto, 2006). The Lü system was first proposed by Lü and Chen (2002) as a model of a nonlinear electrical circuit, and it was studied from various points of view. We intend to study it from mechanical geometry point of view and to point out some of its geometrical and dynamical properties.

In this paper some methods for obtaining the Lagrange functions are pointed out. These methods can be used in general dynamical systems but they are especially useful in the study of autonomous systems. The obtained Lagrange functions is used in dynamical systems with applications in mechanics.

The paper includes the stability study of some dynamical systems given by systems of differential equations. The paper examines the stability of three dynamic systems using the Leapunov function method. The originality of the paper consists of how we choose the Leapunov function. We apply the stability theorems given by Leapunov for autonomous systems. Stability is an important property of a dynamic system that has applications in the technique.

In this paper, starting from recent data provided by the National Institute of Statistics, we analyze the tourism activity in Mehedinți County. We apply the regression method and analyze some models. We also compare the economic results with those of previous years. Also, in this paper we study the stability of dynamic systems with applications in economics. The stability study is done using the Leapunov function method. The originality of the paper consists in the way we choose the mathematical model in case of regression and in the way we choose the Leapunov function in case of dynamic systems in which we analyze stability.