Conjugate Indices

Definition: If $p > 1$ then the Conjugate Index of $p$ is the number $q > 1$ such that $\displaystyle{\frac{1}{p} + \frac{1}{q} = 1}$. If $p = 1$ we define its conjugate index to be $q = \infty$.

For example, consider $p = 3$. Then the conjugate index of $p$ is $q = \frac{3}{2}$ since:

(1)

\begin{align} \quad \frac{1}{p} + \frac{1}{q} = \frac{1}{3} + \frac{1}{\frac{3}{2}} = \frac{1}{3} + \frac{2}{3} = 1 \end{align}

Note that if $p > 1$ and $p \neq \infty$ then we can obtain an explicit formula for $q$:

(2)

\begin{align} \quad \frac{1}{p} + \frac{1}{q} &= 1 \\ \quad \frac{1}{q} &= 1 - \frac{1}{p} \\ \quad \frac{1}{q} &= \frac{p - 1}{p} \\ \quad q &= \frac{p}{p - 1} \end{align}