Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some special cases of more general things. That is called connection.
Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities


You can search on internet with keywords connections matrix inequalities.