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Recently, Garvan obtained two-variable Hecke-Rogers identities for threeuniversal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basichypergeometric functions, and he proposed a problem of finding direct proofs ofthese identities by using Bailey pair technology. In this paper, we give proofsof Garvan's identities by applying Bailey's transform with the conjugate Baileypair 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 oftwo-variable Hecke-Rogers identity related to $g_3(z;q)$, which imply thecorresponding identity given by Garvan. We also extend these two-variableHecke-Rogers identities into infinite families.

The Bailey transform and Hecke-Rogers identities for the universal mock theta functions