Renormalization group analyses of the Yukawa and the quartic scalar couplings are made at the one-loop level in the framework of grand unified theories. The standard Weinberg-Salam model with one Higgs scalar doublet and that with two massless Higgs scalar doublets are examined numerically. The particle mass bounds, the fermion mass ratio and the behavior of the quark mixing angles are studied. In the latter model, some discrete symmetries are assumed in order to avoid flavor nonconserving neutral scalar interactions.