The present thesis investigates higher monotonicity properties in function theory. It consists of three manuscripts. The first one focuses in the beta distribution and its quantiles. It proves logarithmic concavity of the quantiles with respect to the first parameters. The second manuscript computes asymptotic expansions for the quantiles for the first parameter going to 0 or to infinity. The third manuscript is a generalisation of a complete monotonicity result on ratios of gamma functions to entire functions. It also gives a different point of view to previously known results, which were shown only using dedicated properties of the gamma and digamma functions.