It does not seem that your expression has a definite sign:
inst = FindInstance[
100 > qH > qL > b > p > 0 && 0 < \[Rho] < 1 && 2 > \[Lambda] > 0 &&
0 < \[Alpha] < 1 && 2 > m > 1, {m, qH, qL, b,
p, \[Rho], \[Lambda], \[Alpha]}, Reals, 3];
expr = 1 + (b^
m E^((qL \[Rho])/\[Lambda]) m p^(-1 + m) RqL (E^(b/\[Lambda]) +
E^((qL \[Rho])/\[Lambda]) (-1 + \[Alpha]) -
E^((b^m qH)/((b^m + p^m) \[Lambda])) \[Alpha]))/((b^m +
p^m)^2 (-E^(((2 b)/\[Lambda])) +
E^((b + (b^m qH)/(b^m + p^m))/\[Lambda]) +
E^((b^m (qH + qL))/((b^m + p^m) \[Lambda])) (-1 + \[Alpha]) -
E^((2 qL \[Rho])/\[Lambda]) (-1 + \[Alpha]))) + (b^(2 m) \
E^((qL \[Rho])/\[Lambda]) m p^(-1 +
m) RqL (E^((b^m (qH + 2 qL))/((b^m + p^m) \[Lambda])) (qH -
qL) (-1 + \[Alpha]) +
E^(b/\[Lambda]) (-E^(((2 b)/\[Lambda])) qL -
2 E^((b^m (qH + qL))/((b^m + p^m) \[Lambda])) (qH -
qL) (-1 + \[Alpha]) -
2 E^((b + (b^m qL)/(b^m +
p^m))/\[Lambda]) qL (-1 + \[Alpha]) +
E^((2 qL \[Rho])/\[Lambda]) qL (-1 + \[Alpha]) -
E^((2 b^m qH)/((b^m + p^m) \[Lambda])) qL \[Alpha] +
E^((b + (b^m qH)/(b^m + p^m))/\[Lambda]) (-qH +
qL + (qH + qL) \[Alpha]))))/((b^m +
p^m)^3 (E^((2 b)/\[Lambda]) -
E^((b + (b^m qH)/(b^m + p^m))/\[Lambda]) -
E^((b^m (qH + qL))/((b^m + p^m) \[Lambda])) (-1 + \[Alpha]) +
E^((2 qL \[Rho])/\[Lambda]) (-1 + \[Alpha]))^2 \[Lambda]);
Plot[expr /. inst[[2]], {RqL, -1, 1}]