The best bounds in Gautschi's inequality

Different approach to both Gautschi's inequalities (1) and (2) is given. This results in obtaining the best upper bound in (1) and the best lower bound in (2). The main result is the proof of the convexity of the function [(x+t)/(x+s)] 1/(t-s) for |t-s| < 1 . Several new very simple inequalities for digamma function, like (x) < exp(- (x)) or (x + 1) < log(x + e - ) are also proved.