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Given that $p_k> 0$ and $p_1+p_2+\cdots+p_n=1$, prove that \begin{equation} \sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2\geq n^3+2n+\frac{1}{n}. \end{equation}

I believe that Cauchy's inequality should be used at some point but I haven't figured out how. Expanding the square on the left-hand side gives $2n$ immediately, but I have problem producing the cubic term and $\frac{1}{n}$. Could anyone please offer some insight? It is ok to use any method, not necessarily Cauchy's inequality.

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    The minimum of the left hand side occurs when all $p_k$ are equal so $p_k = 1/n$ and the sum becomes $n\cdot(n + 1/n)^2 = n^3 + 2n + \frac{1}{n}$. – 2017-01-17
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    Are you allowed to use Jensen's Inequality instead of Cauchy's Inequality? – 2017-01-17
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    @JimmyK4542 Yes. Any method should be ok. – 2017-01-17
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    @Winther Good point! Could you please suggest a way to justify that minimality can be achieved by forcing all $p_k$ equal? – 2017-01-17
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    We can 'justify' it from it being a symmetric problem and symmetric problems usually have symmetric solutions. As for proving it, a common way to do it is to use the powerful (but maybe not so elegant) method of Lagrange multipliers. – 2017-01-17
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    $p=3, q=\frac {3}{2}, x_k=(p_k)^{\frac {2}{3}}, y_k=(p_k)^{-\frac {2}{3}}$ Answer to the comment on my answer. – 2017-01-18
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    @Winther Nice explanation. Thank you very much! – 2017-01-19
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    @S.C.B. Thank you very much! – 2017-01-19

4 Answers 4

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Let $f(x) = \left(x+\dfrac{1}{x}\right)^2 = x^2+2+\dfrac{1}{x^2}$. Then, $f''(x) = 2+\dfrac{6}{x^4} > 0$ for all $x \in [0,1]$.

Since $f$ is convex on $[0,1]$, for any $p_1,\ldots,p_n \in [0,1]$ where $p_1+\cdots+p_n = 1$, we can use Jensen's Inequality to get $$\dfrac{1}{n}\sum_{k = 1}^{n}f(p_k) \ge f\left(\dfrac{1}{n}\sum_{k = 1}^{n}p_k\right)$$ $$\dfrac{1}{n}\sum_{k = 1}^{n}\left(p_k+\dfrac{1}{p_k}\right)^2 \ge f\left(\dfrac{1}{n}\right)$$ $$\sum_{k = 1}^{n}\left(p_k+\dfrac{1}{p_k}\right)^2 \ge nf\left(\dfrac{1}{n}\right) = n^3+2n+\dfrac{1}{n}.$$

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Note that $$\sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2 =\sum_{k=1}^n p_k^2+2n+\sum_{k=1}^n \frac{1}{p_k^2}$$

Now note that $$\left(\sum_{k=1}^n p_k^2\right)\left(\sum_{k=1}^n 1\right) \ge \left(\sum_{k=1}^n p_k\right)^2 \implies \sum_{k=1}^n p_k^2 \ge \frac{1}{n} $$$$\left(\sum_{k=1}^n \frac{1}{p_k^2}\right)\left(\sum_{k=1}^n p_k\right)\left(\sum_{k=1}^n p_k\right) \ge \left(\sum_{k=1}^n 1\right)^3 \implies \sum_{k=1}^n \frac{1}{p_k^2} \ge n^3$$

From Hölder's inequality.

Adding these two inequalities, we are done.

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    https://hcmop.wordpress.com/2012/04/19/holders-inequality/ – 2017-01-17
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    Could you please point out the choice of $p$ and $q$ in Hölder's inequality? Thank you. – 2017-01-18
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$\sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2 = \sum_{k=1}^n \left(p_k^2 + \frac{1}{p_k^2}+2\right)=\sum_{k=1}^n p_k^2 + \sum_{k=1}^n \frac{1}{p_k^2}+2n\,$. By AM-GM:

  • $\;\;\sqrt{\cfrac{1}{n}\sum_{k=1}^n p_k^2} \;\ge\; \cfrac{1}{n} \sum_{k=1}^n p_k = \cfrac{1}{n} \quad\implies\quad \sum_{k=1}^n p_k^2 \;\ge\; \cfrac{1}{n}$

  • $\;\;\sqrt{\cfrac{n}{\sum_{k=1}^n \cfrac{1}{p_k^2}}} \;\le\; \cfrac{1}{n} \sum_{k=1}^n p_k = \cfrac{1}{n} \quad\implies\quad \sum_{k=1}^n \cfrac{1}{p_k^2} \ge n^3$

Adding up the above gives the stated inequality and, since it's all based on AM-GM, the equality holds iff all $p_k$ are equal.

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The question is from chapter 1 of J. Michael Steele's book ``The Cauchy-Schwarz Master Class''. I didn't know that the book has solutions at the back. Below is the author's solution, which only employs Cauchy's inequality.

First, we expand the square. \begin{equation} \sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2 = \sum_{k=1}^n p_k^2 + 2n + \sum_{k=1}^n \frac{1}{p_k^2}. \end{equation}

Consider the first sum. Notice that \begin{equation} \left(\sum_{k=1}^n p_k^2 \right) \left(\sum_{k=1}^n 1\right) \geq \left(\sum_{k=1}^n p_k\right)^2=1, \end{equation} from which we obtain \begin{equation} \sum_{k=1}^n p_k^2 \geq \frac{1}{n}. \end{equation}

Now, consider the second sum, for which we first consider the sum $\sum_{k=1}^n\frac{1}{p_k}$. Observe that \begin{equation} \sum_{k=1}^n 1 = \sum_{k=1}^n \sqrt{p_k}\frac{1}{\sqrt{p_k}} \leq \left(\sum_{k=1}^n p_k\right)^{\frac{1}{2}} \left(\sum_{k=1}^n \frac{1}{p_k} \right)^{\frac{1}{2}}. \end{equation} It follows that \begin{equation} \sum_{k=1}^n \frac{1}{p_k} \geq n^2. \end{equation}

Next, we can obtain a lower bound for $\sum_{k=1}^n\frac{1}{p_k^2}$. By Cauchy's inequality, \begin{equation} \sum_{k=1}^n\frac{1}{p_k} = \sum_{k=1}^n\frac{1}{p_k} 1 \leq \left(\sum_{k=1}^n\frac{1}{p_k^2}\right)^{\frac{1}{2}} \left(\sum_{k=1}^n 1^2\right)^{\frac{1}{2}}. \end{equation} Hence, \begin{equation} \left(\sum_{k=1}^n\frac{1}{p_k^2}\right)^{\frac{1}{2}} \left(\sum_{k=1}^n 1^2\right)^{\frac{1}{2}} \geq \sum_{k=1}^n\frac{1}{p_k} \geq n^2. \end{equation} Squaring both sides, we obtain \begin{equation} \left(\sum_{k=1}^n\frac{1}{p_k^2}\right) n \geq n^4. \end{equation} Therefore, \begin{equation} \sum_{k=1}^n\frac{1}{p_k^2} \geq n^3. \end{equation}

The conclusion that \begin{equation} \sum_{k=1}^n \left(p_k+\frac{1}{p_k}\right)^2 \geq n^3 + 2n + \frac{1}{n} \end{equation} follows.