The setup: Suppose that a random variable $X$ is measured $n$ times by conducting a physical experiment. The experimenter finds the mean $\overline{X}$, then throws out all other data (and has no memory of what the distribution of $X$ might look like). Now, how can the mean of $T = f(X)$ be estimated using just this information? Here, $f(X)$ is a nonlinear function, assume any other properties if needed like smoothness.
For example, say that $$ T = \sqrt{\frac{7}{X}} $$ and its found that after conducting 50 experiments, $\overline{X} = 10$. Based on this, what is the best estimate of $\overline{T}$? It both feels like it should and shouldn't be $$ \overline{T} = \sqrt{\frac{7}{10}}. $$
My thoughts: first, this problem purposely isn't rigorous, the word "best" is up to your interpretation but should be reasonable. The expectation of a nonlinear function isn't in general the nonlinear function of the expectation, $E[f(X)] \neq f(E[X])$ in general. On the other hand, it feels intuitively correct that the average of $T$ would be $\sqrt{0.7}$.
There is really nothing you can say without making additional assumptions. As a concrete example, let us assume that $X$ is sampled from a normal distribution with a known mean $0$ and an unknown variance, which we denote by $\sigma^2$. Take $f(x)=x^2$, so that $Ef(X)=\sigma^2$. We want to find some continuous function $F$ such that $F(\overline{X})\to \sigma^2$ as $n\to\infty$ (for definiteness, let us say the convergence should be in probability, although this is not so important for now). The crucial point is that $F$ cannot depend on $\sigma$, because $\sigma$ is not known to us; rather we must pick one $F$ that works for any $\sigma$.
However, $\overline{X}\to 0$ almost surely by the strong law of large numbers, so $F(\overline{X})\to F(0)$. Thus as we collect more and more data, the estimator will just converge to some constant value independent of the true value of $\sigma$.