The lateral density distributions (LDD) of inclined cosmic ray air shower are asymmetric and the corresponding iso-density contours are of increasing eccentric ellipses with zenith angles of different showers. The polar asymmetry of the iso-density contours introduces a significant shift of the EAS core, which is quantitatively expressed as a gap length (GL) parameter between the EAS core and the center of the modified density pattern consisting of several equi-density ellipses. The LDD of EAS particles is usually approximated by a particular type of lateral density function (LDF) which is generally assumed to be polar symmetric about the EAS axis, and cannot describe the asymmetric LDDs accurately. A polar angle-dependent modified lateral density function of EASs has been derived analytically by considering the effect of attenuation of EAS particles in the atmosphere. From the simulation studies, it has been found that the GL manifests sensitivity to the cosmic ray mass composition. The cosmic ray mass sensitivity of the lateral shower age is also re-examined by applying the modified LDF to the simulated data.