Constant-roll inflation in scalar-tensor gravity

We generalize the notion of constant-roll inflation earlier introduced in General Relativity (GR) and $f(R)$ gravity to inflationary models in more general scalar-tensor gravity. A number of novel exact analytic solutions for a FLRW spatially flat cosmological background is found for this case. All forms of the scalar field potential and its coupling to gravity producing the exact de Sitter solution, while the scalar field is varying, are presented. In the particular cases of induced gravity and GR with a non-minimally coupled scalar field, all constant-roll inflationary solutions are found. In the former case they represent power-law inflation, while in the latter case the solution is novel and more complicated. Comparison of scalar perturbations generated during such inflation in induced gravity with observational data shows that the constant-roll parameter should be small, similar to constant-roll inflation in GR and $f(R)$ gravity. Then the solution reduces to the standard slow-roll one with small corrections.