Sharpness of Muqattash-Yahdi problem

let ψ denote the psi (or digamma) function. we determine the values of theparameters p, q and r such that ψ(n) ≈ ln(n + p) - is the best approximations. also, we present closer bounds for psi function, which sharpens some known results due to muqattash and yahdi, qi and guo, and mortici. mathematical subject classification: 33b15, 26d15.