  From a combinatorial perspective, we establish three inequalities on
coefficients of $R$- and Kazhdan-Lusztig polynomials for crystallographic
Coxeter groups: (1) Nonnegativity of $(q-1)$-coefficients of $R$-polynomials,
(2) a new criterion of rational singularities of Bruhat intervals by sum of
quadratic coefficients of $R$-polynomials, (3) existence of a certain strict
inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is
to understand Deodhar's inequality in a connection with a sum of
$R$-polynomials and edges of Bruhat graphs.
