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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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