In 1938, Gini studied a mean having two parameters. Later, many authors studied properties of this mean. In particular, it contains the famous means as harmonic, geometric, arithmetic, etc. Here we considered a sequence of inequalities arising due to particular values of each parameter of Gini's mean. This sequence generates many nonnegative differences. Not all of them are convex. We have studied here convexity of these differences and again established new sequences of inequalities of these differences. Considering in terms of probability distributions these differences, we have made connections with some of well known divergence measures.