(restricted to the unit disc) is defined on the unit circle by If Pr denotes the Poisson kernel, it follows from the subharmonicity that It can be shown that the last integral is less than the value at e of the Hardy–Littlewood maximal function φ∗ of the restriction of φ to the unit circle T, so that 0 ≤ M φ ≤ φ∗. It is known that the Hardy–Littlewood operator is bounded on Lp(T) when 1 < p < ∞. It follows that for some universal constant C, If f is a function holomorphic in Ω and 0 < p < ∞, then the preceding inequality applies to φ = |f|. It can be deduced from these facts that any function F in the classical Hardy space Hp satisfies With more work, it can be shown that F has radial limits F(e) almost everywhere on the unit circle, and (by the dominated convergence theorem) that Fr, defined by Fr(e) = F(re) tends to F in Lp(T). Subharmonic functions on Riemannian manifolds Subharmonic functions can be defined on an arbitrary Riemannian manifold. Definition: Let M be a Riemannian manifold, and an upper semicontinuous function. Assume that for any open subset , and any harmonic function f1 on U, such that on the boundary of U, the inequality holds on all U. Then f is called subharmonic. This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality , where is the usual Laplacian. See also Plurisubharmonic function — generalization to several complex variables Classical fine topology Notes References Potential theory Complex analysis Types of functions