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Prove that for all $n \in \mathbb{N}$ the inequality $$\sum \limits_{k=1}^n \frac{k+1}{k} \cdot \sum \limits_{k=1}^n \frac{k}{k+1} \le \frac{9}{8}n^2$$ holds.

My work. I proved this inequality, but my proof is ugly (it is necessary to check by brute force whether the inequality holds for $n=1,2,3,4,5$).

Witold
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2 Answers2

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The first inequality is equivalent to $$(n+H_n)(n+1-H_{n+1})\le\frac98n^2$$ where $H_n$ denotes the $n$th Harmonic number. It's well known that $H_n=\gamma+\ln{(n)}+o(1)$ as $n\to\infty$ so it's clear that the LHS is asymptotic to $n^2+n+o(n)$. In particular, we have that the LHS divided by $n^2$ is decreasing. So we just need to find the ratio of this function and $n^2$ for small $n$ and as the ratio begins to decrease, it will not increase again due to the function's asymptotic behaviour. Using this gives us the much stricter inequality $$(n+H_n)(n+1-H_{n+1})\le\frac{104273}{100800}n^2$$ for $n\in\mathbb{N}$. Equality only occurs when $n=6$ and the ratio is at a maximum.

The second inequality is equivalent to $$(n^2+2n-H_n)(n^2-2n-3+3H_{n+1})\le\frac98n^4$$ A similar analysis to before gives us the asymptotic $n^4+n^2(2\ln{(n)}+2\gamma-7)+o(n^2)$ for the LHS. Dividing this by $n^4$ gives a decreasing ratio so we again only need to analyse small values of $n$. This gives us the stricter inequality $$(n^2+2n-H_n)(n^2-2n-3+3H_{n+1})\le\frac{1975}{1944}n^4$$ for $n\in\mathbb{N}$. Equality only occurs when $n=3$ in this case.

J. W. Tanner
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Peter Foreman
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First problem.

Let $x_k=\frac{k+1}{k}.$

Thus, $x_k\in[1,2]$ for all $k$.

Now, let $$f(x_1,x_2,...x_n)=\sum_{k=1}^nx_k\sum_{k=1}^n\frac{1}{x_k}.$$ Easy to see that $f$ is a convex function of $x_k$ for all $k$.

Thus, by AM-GM $$\max_{x_k\in[1,2]}f=\max_{x_k\in\{1,2\}}f=\max_{i+j=n}(i+2j)\left(i+\frac{j}{2}\right)=$$ $$=\max_{i+j=n}\left((i+j)^2+\frac{1}{2}ij\right)\leq n^2+\frac{1}{2}\left(\frac{n}{2}\right)^2=\frac{9}{8}n^2.$$

Michael Rozenberg
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  • I'm very interested in your proof, but unfortunately I do not understand your proof. Please explain this step $$\max_{x_k\in[1,2]}f=\max_{x_k\in{1,2}}f=\max_{i+j=n}(i+2j)\left(i+\frac{j}{2}\right)$$ in more detail. Also i don't know what is AM-GM. – Witold Aug 03 '19 at 09:00
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    @Witold Convex function gets a maximal value for extreme value of the variable. Draw it! For example if you'll take a parabola $y=x^2$ on $[-1,2]$ so since $y=x^2$ is a convex function $x^2$ gets a maximal value for $x=-1$ or for $x=2$, which gives $\max_{[-1,2]}x^2=\max{(-1)^2,2^2}=4.$ In our case it's enough to prove our inequality for $x_k\in{1,2}$. AM-GM here it's the following: $ij\leq\left(\frac{i+j}{2}\right)^2.$ – Michael Rozenberg Aug 03 '19 at 09:09
  • Thank you! I understand what you wrote. But how to prove that the function $f$ is convex? – Witold Aug 03 '19 at 09:40
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    @Witold I did not say that $f$ is a convex function. I said that $f$ is a convex function of $x_k$ for all $k$. I see it so. Let $x_k=x$. Thus, $f=(x+A)\left(\frac{1}{x}+B\right)=1+AB+Bx+\frac{A}{x},$ which is a convex function of $x$. – Michael Rozenberg Aug 03 '19 at 09:46
  • Very nice proof! Thank you very much! – Witold Aug 03 '19 at 09:55
  • @Witold You are welcome! Good luck! – Michael Rozenberg Aug 03 '19 at 09:59