Given $X_1,...,X_n$ ($n\geq 2$) are iid and each have density:
$f_X(x) = \frac{c^\theta \theta}{x^{1+\theta}}\mathbb{1}(x> c)$ for known $c$ and $\theta > 1$
then we can easily find the first moment which is $E(X)=\mu$ given by the integral $\int_c^\infty x\frac{c^\theta \theta}{x^{1+\theta}} dx$. This value is $\mu = \frac{c\theta^n}{\theta - 1}$.
The likelihood function is
$\mathcal{L}(x;\theta) = \frac{c^{n\theta}\theta^n}{\left(\prod_{i=1}^n x_i\right)^{1+\theta}}\mathbb{1}(\min X > c) = \frac{c^{n\theta}\theta^n}{\exp\left((1+\theta)\sum_{i=1}^n \log x_i\right)}\mathbb{1}(\min X > c)$
Now suppose we wanted to test $H_0: \mu = \mu_0$ against $H_a: \mu > \mu_0$ for some constant $\mu_0$ of interest. I can show that $\mathcal{L}$ is decreasing on $T(X) = \sum_{i=1}^n \log x_i$ (in fact it is minimally sufficient for $\theta$), so we can get a monotone likelihood ratio and thus was have a UMP test exists of the form $T(X) > k$ where $k$ is chosen to satisfy a Type I error of $\alpha$.
However, this appears to be a test for $\theta = \theta_0$ and $\theta > \theta_0$, and not $\mu = \frac{c\theta^n}{\theta - 1}$. I am unsure how to proceed with finding a UMP test for $\mu = \mu_0$. It doesn't seem that we can write $\theta$ as a function of $\mu$.
On one hand, $\mu$ is not monotonic on $\theta > 1$, so we would also worry about $T(X) > k$ or $T(X) < k$ being a test for $\mu$. I do not know how to express the the test for $\mu$. In general, is there some way to form a UMP test for $H_0: \mu = \mu_0$ and $H_a: \mu > \mu_0$ for each possible function of $\theta$ given by $g(\theta)$?
Update:
I made a mistake in $E(X) = \int_c^\infty x\frac{c^\theta \theta}{x^{1+\theta}} dx$. This value is $\mu = \frac{c\theta^n}{\theta - 1}$. There should be no $n$ in the exponent of $\theta$. Rather $E(X) = \frac{c \theta}{1+\theta} \equiv \mu$
Then we can solve for $\mu(\theta) = \frac{\mu}{\mu - c}$, $\mu > c$. The writing the likelihood as a function of $\mu$,
$\mathcal{L}(\mu;x) = \frac{c^{\frac{n\mu}{\mu-c}}\left(\frac{\mu}{\mu-c}\right)^n}{\left(\prod_{i=1}^n x_i\right)^{\frac{2n\mu - cn}{\mu - c}}}\mathbb{1}(\min X > c)$.
Since $\mathcal{L}$ is decreasing on $\prod_{i=1}^n x_i$, we can use the monotone likelihood ratio to assert there is a UMP test which rejects when $\prod_{i=1}^n x < k$, where $k$ is chosen to satisfy the type I error rate $\alpha$.
Is this approach correct?
