H[t_] = Log[t] - (t - 1)/(λ t + μ)
The definition domain and the parameter ranges for the function H are:
0 < t < 1, μ >= λ > 0
But the range of the function H cannot be determined directly using the following code:
H[t_] = Log[t] - (t - 1)/(λ t + μ)
FunctionRange[{H[t], 0 < t < 1, μ >= λ > 0}, t, y]