Is the product of two subharmonic function necessarily subharmonic?

Question

Defin:

A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}{{2\pi }}\int_0^{2\pi } {\varphi ({\lambda _0} + r{e^{i\theta }})d\theta } $$

Now let $\varphi $ and $ \Gamma $ are subharmonic.

Can we say that $\varphi ({\lambda _0})\Gamma ({\lambda _0})$ is subharmonic?

Answer 1

Hint: Consider the subharmonic functions $|z|,-1.$