Inequality with polygamma functions

For n=1, 2, \dots and z>0 denote by \psi^{(n)} the n -th polygamma function. Prove that

    \[\psi^{(m)}(z) \psi^{(n)} (z) \geq \psi^{(\frac{m+n}{2})} (z)\]

where \frac{m+n}{2} is an integer.

Solution

We begin by the integral representation of the polygamma function

    \[\psi^{(n)} (z) = (-1)^{n+1} \int_{0}^{\infty} \frac{t^n}{1-e^{-t}} e^{-zt} \, {\rm d}t\]

All we have to apply is this generalization of the Schwartz inequality

\displaystyle \int_{a}^{b} g(t) f^m (t) \, {\rm d}t \int_{a}^{b} g(t) f^n(t) \, {\rm d}t \geq \int_{a}^{b} g(t) f^{(m+n)/2} (t) \, {\rm d}t

So if we choose f(t)=t and g(t)=\frac{e^{-zt}}{1-e^{-t}} and a=0 as well as b=+\infty the result follows immediately.

Remark: As a corollary we get that

    \[\frac{\psi^{(n)}(z)}{\psi^{(n+1)}(z)} \geq \frac{\psi^{(n+1)}(z)}{\psi^{(n+2)}(z)} \quad , \quad z >0\]

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