This paper introduces the theory of φ-Jensen variance. Our main motivation is to extend the connotation of the analysis of variance and facilitate its applications in probability, statistics and higher education. To this end, we first introduce the relevant concepts and properties of the interval function. Next, we study several characteristics of the log-concave function and prove an interesting quasi-log concavity conjecture. Next, we introduce the theory of φ-Jensen variance and study the monotonicity of the interval function JVar φ ϕ(X [a,b] ) by means of the log concavity.Finally, we demonstrate the applications of our results in higher education, show that the hierarchical teaching model is 'normally' better than the traditional teaching model under the appropriate hypotheses, and study the monotonicity of the interval function Var A (X [a,b] ). MSC: 26D15; 62J10

Abstract. By means of the theory of majorization and under the proper hypotheses, the following inequalities of Jensen-Pečarić-Svrtan-Fan (Abbreviated as J-P-S-F) type are established:

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Abstract. The well-known Maclaurin's inequalities are generalized as follows: If x and y are two positive n -tuples, and y and x/y are similarly ordered, thenn (x)/P [1] n (y) P [2] n (x)/P [2] n (y) · · · P [k] n (x)/P [k] n (y) · · · P [n] n (x)/P [n] n (y), where P [k] n (a) is the k -th symmetric mean of a (see [15], p. 283]). The method used in this paper is based on the computational method of descending dimension.

Let p be a real density function defined on a compact subset of R m , and let E(f , p) = pf dω be the expectation of f with respect to the density function p. In this paper, we define a one-parameter extensionof a positive continuous function f defined on . By means of this extension, a two-parameter mean V r,s (f , p), called the Dresher variance mean, is then defined. Their properties are then discussed. In particular, we establish a Dresher variance mean inequality min t∈ {f (t)} ≤ V r,s (f , p) ≤ max t∈ {f (t)}, that is to say, the Dresher variance mean V r,s (f , p) is a true mean of f . We also establish a Dresher-type inequality V r,s (f , p) ≥ V r * ,s * (f , p) under appropriate conditions on r, s, r * , s * ; and finally, acan be compared with E(f , p). We are also able to illustrate the uses of these results in space science. MSC: 26D15; 26E60; 62J10

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.

Let M [t] n (a) be the tth power mean of a sequence a of positive real numbers, where a = (a 1 ,a 2 ,...,a n ),n ≥ 2, and α,λ ∈ R m ++ ,m ≥ 2, m j=1 λ j = 1,min{α} ≤ θ ≤ max {α}. In this paper, we will state the important background and meaning of the inequalityn (a); a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.Copyright © 2006 J. Wen and W.-L. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Symbols and introductionWe will use some symbols in the well-known monographs [1,5,13]:is a finite set, and it is not empty.Recall that the definitions of the tth power mean and Hardy mean of order r for a sequence a = (a 1 ,...,a n ) (n ≥ 2) are, respectively,, if 0< |t| < +∞,n (a) = n √ a 1 a 2 ··· a n , if t = 0, Hindawi Publishing Corporation