Abstract. The weak monotonic function is defined in this paper. We will study the relationship between the weak monotonic function and the Schur-function. We show that a Schur-convex function is a weak increasing function under the proper hypotheses. By means of the theory of weak monotonic function with appropriate assumptions, we have established a Chebyshev type inequality as follows:As the application of the inequality, a new proof of Marshall's inequality is obtained.Mathematics subject classification (2010): 26D15.

Abstract. By means of the analysis, convex geometry, computer and majorization theories, in the centered 2 -surround system S (2) {P,Γ,l} , we establish the following mean central distancecentral distance inequalities:where τ = 2.49342812654089 ... , and τ/2 is the unique real root of the following equation:We also demonstrate the applications of our results, and obtain the N -mean central distancecentral distance inequality and the mean central distance-central distance-limit inequality.Mathematics subject classification (2010): 26D15, 26E60, 51K05, 52A40.

No abstract

In the centered surround system S (2) {P, Γ }, we establish the following gravity inequalities:where Γ is an ellipse, P and e are one of the foci and the eccentricity of the ellipse, respectively, and A ∈ Γ is a satellite of the centered surround system S (2) {P, Γ }. We also demonstrate the applications of the inequalities in the temperature research, and we obtain an approximate mean temperature formula as follows:where the T is the mean temperature on a planet.MSC: 26D15; 26E60; 51K05; 52A40

By means of the mathematical induction, stepwise adjustment method and the reorder method, under the proper hypotheses, we established the following Cater type inequalities involving Cater products:As applications, we solved the problem which proposed by M. Laub, Jerusalem and Israelin under the proper hypotheses, and an l -isoperimetric inequality in the centered n -surround system S (2) {P,Γ,l} is obtained as follows:|Γ| n 2π n .

This paper is motivated by several interesting problems in statistics. We first define the concept of quasi-log concavity, and a conjecture involving quasi-log concavity is proposed. By means of analysis and inequality theories, several interesting results related to the conjecture are obtained; in particular, we prove that log concavity implies quasi-log concavity under proper hypotheses. As applications, we first prove that the probability density function of k-normal distribution is quasi-log concave. Next, we point out the significance of quasi-log concavity in the analysis of variance. Next, we prove that the generalized hierarchical teaching model is usually better than the generalized traditional teaching model. Finally, we demonstrate the applications of our results in the research of the allowance function in the generalized traditional teaching model. MSC: 26D15; 62J10