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 Hp 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.
Read more about this topic: Subharmonic Function
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