  Recently, Garvan obtained two-variable Hecke-Rogers identities for three
universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic
hypergeometric functions, and he proposed a problem of finding direct proofs of
these identities by using Bailey pair technology. In this paper, we give proofs
of Garvan's identities by applying Bailey's transform with the conjugate Bailey
pair of Warnaar and three Bailey pairs deduced from two special cases of
$_6\psi_6$ given by Slater. In particular, we obtain a compact form of
two-variable Hecke-Rogers identity related to $g_3(z;q)$, which imply the
corresponding identity given by Garvan. We also extend these two-variable
Hecke-Rogers identities into infinite families.
