A Pair of Interesting Inequalities

In research with product partitions models, especially the Dirichlet process mixture, I've found a pair of inequalities useful. They are reproduced as theorems below. Does anyone know if these are special cases of a well-known theorem?

Theorem: Let and be positive integers such that . Then,

.

Proof: implies

Theorem: Let and be positive integers such that . Then,

.

Proof: implies