To study the ground state of an axial next-nearest-neighbor Ising chain under transverse field as a function of frustration parameter κ and field strength Γ, we present here two different perturbative analyses. In one, we consider the (known) ground state at κ=0.5 and Γ=0 as the unperturbed state and treat an increase of the field from 0 to Γ coupled with an increase of κ from 0.5 to 0.5+rΓ J as perturbation. The first-order perturbation correction to eigenvalue can be calculated exactly and we could conclude that there are only two phase-transition lines emanating from the point κ=0.5, Γ=0. In the second perturbation scheme, we consider the number of domains of length 1 as the perturbation and obtain the zeroth-order eigenfunction for the perturbed ground state. From the longitudinal spin-spin correlation, we conclude that floating phase exists for small values of transverse field over the entire region intermediate between the ferromagnetic phase and antiphase. © 2007 The American Physical Society.