# Recitation 2

After introducing Triangle Inequality and Arithmetic Geometric Mean Inequality, I briefly discussed a more general way to define means. For any ‘good enough’ function $f$, the mean value of $x, y$ generated by $f$ can be defined as $$f^{-1}\left(\frac{f(x)+f(y)}{2}\right).$$ The identity map gives rise to the arithmetic mean $A$, the natural log the geometric mean $G$, the reciprocal function the harmonic mean $H$, and the square function the root mean square $S$. And all of those mean values are related by the following inequality chains, $$H \leq G\leq A\leq S.$$