Consider two concave functions on the unit interval $[0,1]$ of the form:
$f_{p_1,q_1}(x)=H_b(xp_1+(1-x)q_1)-xH_b(p_1)-(1-x)H_b(q_1)$
$f_{p_2,q_2}(x)=H_b(xp_2+(1-x)q_2)-xH_b(p_2)-(1-x)H_b(q_2)$
where $H_b(.)$ is the binary entropy function and $p_1,p_2,q_1,q_2$ all are in $(0,1)$ and further $p_1>p_2$ and $q_1<q_2$. The question is if $f_{p_1,q_1}(x)=f_{p_2,q_2}(x)$ holds for at most three points in $[0,1]$ for all feasible choices of $(p_1,p_2,q_1,q_2)$? (clearly $x=0,1$ are always solutions and for some choices $p$'s and $q$'s I can produce a third root).