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
Famous quotes containing the words functions, complex and/or plane:
“In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)
“In ordinary speech the words perception and sensation tend to be used interchangeably, but the psychologist distinguishes. Sensations are the items of consciousness—a color, a weight, a texture—that we tend to think of as simple and single. Perceptions are complex affairs that embrace sensation together with other, associated or revived contents of the mind, including emotions.”
—Jacques Barzun (b. 1907)
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)