For a function with this property on a disk that is non-negative over that disk, an upper and lower bound is provided by Harnack's Inequality.  
For a function with this property on a disk that is non-negative over that disk, an upper and lower bound is provided by Harnack's Inequality.  They're not entire or analytic functions, but any function that has this property over the entire plane and is bounded above or below must be constant according to Liouville's Theorem. The Poisson Integral Formula produces a continuous function with this property on a disk given fixed values for the function on the boundary, which is the objective of the Dirichlet Problem. The real and imaginary parts of an analytic function both have this property and are called "conjugates." The divergence of the gradient of these functions is equal to 0.  For 10 points, name these functions, which satisfy Laplace's Equation, and which are described by the same adjective as the series defined as the sum from n = 1 to infinity of one over n.
For a function with this property on a disk that is non-negative over that disk, an upper and lower bound is provided by Harnack's Inequality.  
Harmonic Functions [or Potential functions before it is read; or specific types of harmonic functions, like spherical harmonics; prompt on things like solutions to Laplace's equation]