Axisymmetric frictionless contact problem of the theory of elasticity on indentation of a non-deformable punch into an elastic transversely-isotropic half-space with transverselyisotropic functionally graded coating is considered. Elastic moduli of the coating vary with depth according to arbitrary function. The technique based on integral transformations is used to reduce the problems to the integral equation. Special approximations for the kernel transform is used to obtain analytical solution of the integral equations. The solution is asymptotically exact for both large and small values of geometric parameter of the problem (relative layer thickness). A method of construction the compliance functions is presented for a case of arbitrary axisymmetric normal and tangential loadings.