Upper bound on expectation of truncated random variable
Solution 1:
I guess you shoud use this expression of the maximum : $$\max(x,y) = \frac{|x-y| + x + y}{2}.$$
Then you would get $$\mathbb{E}[Y] = \mu + \frac{\mathbb{E}[|X - \mu|]}{2}.$$
By Hölder's inequality, you get the bound $$\mathbb{E}[Y] \leq \mu + \frac{\sigma}{2}.$$