Properties of Completely Monotone and Bernstein Functions Related to the Shape of Their Measures

We are interested in Bernstein and Lévy measures having certain convexity-type properties. We thoroughly investigate how the shape of these measures results in various properties of the corresponding completely monotone and Bernstein functions. We hope that by understanding these connections one would be able to shed more light on the several conjectures posed in another paper of the authors ["New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures", J. Theoretical Probability (2014), DOI 10.1007/s10959-014-0557-9].