We study the ground states properties of the even-even neutron-deficient Hg and Pb isotopes using deformed relativistic mean field formalism. The ground state structures are investigated by comparing the prolate, oblate and spherical solutions. We observe shape transitions from oblate to prolate and prolate to oblate at A = 1784 and A = 188, respectively, for Hg isotoes. Some low-lying superdeformed configurations are found in the Hg isotoes, and found to be the ground state for ^<180>Hg. Studies of the quadrupole and the hexadecapole moments, density distributions and the single-particle level structure clearly indicate that some of the neutron-deficient isotopes having proton number Z = 82 do not behave as good magic nuclei.