## Subharmonic behaviour of smooth functions.(English)Zbl 0944.31003

The author considers the subharmonic behaviour of smooth functions defined in a proper subdomain $$G$$ of $$\mathbb{R}^n$$. The main result is as follows:
Theorem. Let $$f\in C^2(G)$$. If there are nonnegative constants $$K$$ and $$K_0$$ such that $\|\Delta f(x)\|\leq {{K}\over{r}} \sup\|\nabla f\|+ {{K_0}\over{r^2}} \sup\|f\|,\qquad x\in G,$ where the supremum is taken over the Euclidean balls $$B_r (x)= \{y:\|y-x\|< r\}\subset G$$, then $$\|f\|^p ,0< p < \infty$$ has subharmonic behaviour, i.e. there exists a constant $$C$$, $$1\leq C$$, such that $\|f(x)\|^p \leq {{C}\over{r^n}} \int_{B_r(x)}\|f(y)\|^p dV(y),\text{ for all } x \in G,$ If in addition $$K_0=0$$, then $$\|\nabla f\|^p$$ has subharmonic behaviour.

### MSC:

 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions

### Keywords:

subharmonic behaviour
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