Sign of expectation of a function of a random variable

Question

Let $X$ be a random variable. Does it hold that $\mathbb{E}\left[ X \right]$ has the same sign as $\mathbb{E}\left[ arctan(X) \right]$? If not, are there any conditions I can impose on the distribution of $X$ for it to hold?

Answer 1

Counterexample:

Let $X$ take values $-1,\sqrt{3}$ with probabilities $0.6,0.4$ respectively. $$E[X]=0.6(-1)+0.4(\sqrt{3})\approx 0.09>0\\ E[\arctan(X)]=0.6(-\pi/4)+0.4(\pi/3)=-\pi/60<0.$$

As for your second question, since you have not specified any necessary restrictions on the distribution of $X$, choose any $X$ that is almost surely positive.