Estimating gamma function by digamma function

The aim of this paper is to refine some recent results stated by Alzer and Grinshpan [H. Alzer, A.Z. Grinshpan, Inequalities for the gamma and qq-gamma functions J. Approx. Theory 144 (2007) 67–83] and Batir [N. Batir, An interesting double inequality for Euler’s gamma function J. Inequal. Pure Appl. Math. 5(4) (2004) article 97] concerning the problem of approximation of the gamma function in terms of the digamma function.