$f(y)\le f(x)$ and $x
Then $(1-\lambda)g(\lambda f(x)+(1-\lambda)f(y)) \le (1-\lambda) g(f(y))$.
I can't understand well.
I guess, since $g$ is increasing function,
the inequality should be $(1-\lambda)g(\lambda f(x)+(1-\lambda)f(y)) \le (1-\lambda) g(f(x))$ because $f(y)\le f(x)$.
Why does that inequality hold?