Exercise
I'm currently reading through Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range, and I've been tasked with the following (Exercise II.4.4):
Let $u$ be subharmonic on $D\subseteq\mathbb{C}$. Show that $u^p$ is subharmonic for $p\geq 1$.
There is also a hint:
Hint: Use Hölder's inequality.
My Thoughts
I feel like I can prove this if I'm given that $u$ is nonnegative on $D$, since Hölder's inequality shows $$ \int_{0}^{2\pi}|u(z_0+\rho e^{it})|\,dt\leq\left(\int_{0}^{2\pi}|u(z_0+\rho e^{it})|^{p}\,dt\right)^{1/p} $$ for some fixed $z_0\in D$ and $\rho>0$ small enough that $B(z_0,\rho)\subseteq D$, and if I'm given $u\geq 0$ on $D$, then $|u|=u$ and combining the above equation with the sub-mean-value property of $u$ should show the result.
Also I feel like if I'm merely given $u$ continuous I could do some local arguments of this sort to show the result, but I'm not even given that.
I had a moment of weakness and looked on the internet to see I could find this proven somewhere, but every source I could find only proved it given $u\geq 0$ on the domain. This makes think that the statement might even be true unless given this extra hypothesis.
Any help is greatly appreciated. Thank you.
It doesn't really make sense to allow $u$ to attain negative values. What is $(-1)^{5/2}$ or $(-1)^{\pi}$ supposed to mean? (Subharmonic functions are real-valued.)
Most likely the author forgot to add the assumption that $u\ge 0$.