Let（设） f be a function with a positive second derivative（二阶导数）. Such a function is called “convex"（凸的，注意是向下凸，向上凹，如 x2 ） and satisfies the inequality
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inequation (1) relates a function of an average to an average of the function. The average can be weighted, for example,
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Figure 1 is a graphical interpretation of inequation (2) for the function f=x2.
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There is nothing special about f=x2, except that it is convex. Given three numbers a, b, and c, the inequality (2) can first be applied to a and b, and then to c and the average of a and b. Thus, recursively, an inequality like (2) can be built for a weighted average（加权平均数） of three or more numbers. Define weights that are normalized（标准化） (
). The general result（通式） is
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If all the pj are the same, then both of the two terms in S are the same, and S vanishes. Hence, minimizing S is like urging all the pj to be identical（完全一样）. Equilibrium is when S is reduced to the smallest possible value which satisfies any constraints that may be applicable. The function S defined by (3) is like the entropy（熵） defined in thermodynamics（热力学）.