The proof of Muqattash–Yahdi conjecture

In this paper we solve an open problem posed by Muqattash and Yahdi [I. Muqattash, M. Yahdi, Infinite family of approximations of the digamma function, Math. Comput. Modelling 43 (2006) 1329–1336], also rediscussed by Qi and Guo [F. Qi, B.-N. Guo, Sharp inequalities for the psi function and harmonic numbers, arXiv:0902.2524v1 [math.CA]].