continuous in the usual topology, however one can introduce the fine topology which makes them continuous. Examples If is analytic then is subharmonic. More examples can be constructed by using the properties listed above, by taking maxima, convex combinations and limits. In dimension 1, all subharmonic functions can be obtained in this way. Riesz Representation Theorem If is subharmonic in a region , in Euclidean space of dimension , is harmonic in , and , then is called a harmonic majorant of . If a harmonic majorant exists, then there exists the least harmonic majorant, and while in dimension 2, where is the least harmonic majorant, and is a Borel measure in . This is called the Riesz representation theorem. Subharmonic functions in the complex plane Subharmonic functions are of a particular importance in complex analysis, where they are intimately connected to holomorphic functions. One can show that a real-valued, continuous function of a complex variable (that is, of two real variables) defined on a set is subharmonic if and only if for any closed disc of center and radius one has Intuitively, this means that a subharmonic function is at any point no greater than the average of the values in a circle around that point, a fact which can be used to derive the maximum principle. If is a holomorphic function, then is a subharmonic function if we define the value of at the zeros of to be −∞. It follows that is subharmonic for every α > 0. This observation plays a role in the theory of Hardy spaces, especially for the study of H when 0 < p < 1. In the context of the complex plane, the connection to the convex functions can be realized as well by the fact that a subharmonic function on a domain that is constant in the imaginary direction is convex in the real direction and vice versa. Harmonic majorants of subharmonic functions If is subharmonic in a region of the complex plane, and is harmonic on , then is a harmonic majorant of in if in . Such an inequality can be viewed as a growth condition on . Subharmonic functions in the unit disc. Radial maximal function Let φ be subharmonic, continuous and non-negative in an open subset Ω of the complex plane containing the closed unit disc D(0, 1). The radial maximal function for the function φ