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 $n_1$ and $n_2$ be positive integers such that $n_1 \geq n_2$. Then,

$${(n_1+1)!(n_2-1)! > (n_1)!(n_2)!}$$

Proof: $n_1 \geq n_2 \geq 1$ implies

$$\begin{array}{r c l} n_1+1 & > & n_2 \\ {(n_1+1)!}/{n_1!} & > & {n_2!}/{(n_2-1)!} \\ {(n_1+1)!(n_2-1)!} & > & {n_1!n_2!} \end{array}$$

Theorem: Let $n_1$ and $n_2$ be positive integers such that $n_1 \geq n_2$. Then,

$$(n_1+1)(n_2-1) < n_1n_2$$

Proof: $n_1 \geq n_2 \geq 1$ implies

$$\begin{array}{r c l} n_1+1 & > & n_2 \\ n_1+ 1 + n_1n_2 - n_1n_2 & > & n_2 \\ n_1n_2 & > & n_1n_2 - n_1 + n_2 - 1 \\ n_1n_2 & > & (n_1+1)(n_2-1) \end{array}$$