Concave function and average of random variable.

Question

Let $X$ be positive random variable. For a natural number $n$, Let $Y_1 , Y_2, ...Y_n $~$^{i.i.d} X$, and let $a_n=E( \ln(\frac{Y_1 +\cdots+Y_n}{n}))$

Then does $a_n \leq a_{n+1}$ hold true? I think this problem with Jensen inequality, but I can't prove it...