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In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the ''boundary'' of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball. ''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions. ==Formal definition== Formally, the definition can be stated as follows. Let be a subset of the Euclidean space be an upper semi-continuous function. Then, is called ''subharmonic'' if for any closed ball of center and radius contained in and every real-valued continuous function on that is harmonic in and satisfies for all on the boundary of we have for all Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition. A function is called ''superharmonic'' if is subharmonic. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「subharmonic function」の詳細全文を読む スポンサード リンク
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