Iosif Pinelis published a paper, “Characteristic function of the positive part of a random variable and related results, with applications,” in Statistics and Probability Letters, Vol. 106 (2015), pp. 281-286.
Professor Iosif Pinelis (Math) published Exact Upper and Lower Bounds on The Difference Between The Arithmetic and Geometric Means, on Cambridge Journals Online Bulletin of the Australian Mathematical Society, first view;
You can read the paper.