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If $f$ is harmonic on an open set then the maximum and minimum principle ($M$) applys to $f$. Are there non trivial known properties $A(f)$ s.t. following holds:

Let $U\subset\mathbb R^n$ open and $f:U\rightarrow\mathbb R$ then $A(f)\wedge M(f)\Leftrightarrow f$ is harmonic.

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Nothing interesting in this direction. The maximum and minimum principles are so much weaker than harmonicity that one would need $A(f)$ to encapsulate harmonicity.

E.g., the composition of a harmonic function with any homeomorphism of $\mathbb{R}^n$ satisfies the maximum and minimum principles, but can be very far from being harmonic. What else would we need this function to satisfy in order to conclude it's harmonic... uh, the Laplace equation, probably.

There are nontrivial results of the form "what should be added to the mean value property to get harmonicity".

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    Thanks for the answer! I've just read now that any continuous function which satisfies the mean value property is smooth. Would you know a reference for that? – 2017-01-16