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An inequality involving harmonic number

Let \mathcal{H}_n denote the n – th harmonic number and let  n \in \mathbb{N}. Prove that

\displaystyle \int_{0}^{1} \frac{{\rm d}x}{x^{n-1} + x^{n-2} + \cdots+x+1} \geq \frac{1}{\mathcal{H}_n}

Solution

We might begin with the integral representation of the harmonic number, namely the equation:

\displaystyle \mathcal{H}_n = \int_{0}^{1} \left ( 1+x+\cdots+x^{n-1} \right ) \, {\rm d}x = \sum_{k=1}^{n} \frac{1}{k}

So we have to prove the equivelant inequality

\displaystyle \int_{0}^{1}\left ( x^{n-1} + \cdots + x +1 \right ) \int_{0}^{1} \frac{{\rm d}x}{x^{n-1} + \cdots + x +1} \geq 1

and this is obvious using the Cauchy – Schwarz inequality.

The exercise can also be found here .

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