Inequality in hamming distance

Let two functions $f,g :[n]\to \{0,1\},$ then we define $$\delta{(f,g)}=\frac{|\{i\in[n]:f(i)\neq g(i) \}|}{n}$$ is called normalized hamming distance(n.h.d).

For example, $f=1011, g=0101$, I clearly see $\delta(f,g)=\frac{3}{4}.$

Suppose $\epsilon=(0,1],$ we have two n.h.d such that, $\delta(h,g)\leq 0.3\epsilon,\delta(f,g)\gt \epsilon,$ from that is it possible to show $\delta(f,h)\gt 0.7\epsilon,$ by using triangle inequality?