Comment by kqr

You can plot

g = 0.6 log (1 + 2f) + 0.4 log (1 - f)

And locate f=0.5 and compare to the maximum g.

Edit: I wanted to check my intuition so I did: https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B...

Looks like 0.5 is a slight overbet, but still very, very good.