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This paper presents majorization results for PH-generators. Based on the majorization results, bounds on the moments and Laplace-Stieltjes transforms of phase-type distributions are found. Exponential distributions and Coxian distributions are identified to be extremal PH-distributions with respect to all the moments and Laplace-Stieltjes transforms for certain subsets of PH-distributions. UN CO RR EC TE D PR OO F Chapter 6 1 Majorization and Extremal PH Distributions 2 Qi-Ming He, Hanqin Zhang, and Juan C. Vera 3