Greetings and many thanks in advance.
Let $h(p)=-p\log_2 p-\bar{p}\log_2 \bar{p}$ be the binary entropy function, where $\bar{p}=1-p$. Prove that $$(1-h(p))h(x)-h(p*x)+h(p)\geq 0,\forall x\in [0,1],$$ where $p*x=p\bar{x}+x\bar{p}.$
It is trivial to see that we can only check $x\in [0,1/2]$. And also, I plotted the graph of the LHS function, 