Is the Shapiro inequality for $n=23$ an open problem? The reason why I am asking is I have two contradictory pieces of information from two different articles.

The first article titled "The validity of Shapiro’s cyclic inequality" published by B.A. Troesch in the journal "Mathematics of Computation" in 1989 claims that the inequality does indeed holds for all odd $N \leq 23$. The link to the article is here.

However, there is another article published in 2002, titled "Shapiro's Cyclic Inequality for Even n" by Bushell, PJ and McLEOD, JB in the journal "Journal of Inequalities and Applications" that claims that the case $n=23$ is still an open problem. Here is a link to the article. Interestingly, this article cites the previous article.

The Shapiro inequality is the claim that if $x_1,x_2,\ldots,x_n \in \mathbb{R}^+$, then $$\sum_{k=1}^{n} \dfrac{x_k}{x_{k+1} + x_{k+2}} \geq \dfrac{n}2$$ where $x_{n+1} = x_1$ and $x_{n+2} = x_2$ holds true for $n \in \{1,2,3,\ldots,12,13,15,17,19,21,23 \} $.

Here is the wikipedia link for whatever it is worth.

  • $\begingroup$ It is not open. Actually, I think there are non completely computational proofs for n=23 too. $\endgroup$ – Gjergji Zaimi Jul 10 '12 at 10:57
  • $\begingroup$ The author of the following paper claims to settle the case $n=23$ affirmatively---the proof still looks messy, so I did not try to verify it: math.s.chiba-u.ac.jp/~ando/Shapiro.pdf $\endgroup$ – Suvrit Jul 10 '12 at 20:32
  • $\begingroup$ @Gjergji Zaimi : Could you point me to an appropriate reference? $\endgroup$ – user11000 Jul 10 '12 at 20:33
  • $\begingroup$ @Suvrit: Thanks. I actually looked at the paper as well. But I am not completely sure if the author finally proves it or not. As you say, the material gets a bit unclean towards the end and is difficult to follow. $\endgroup$ – user11000 Jul 10 '12 at 20:33

There is no incongruity between the two papers that have been cited in the question.

According to A.M. Fink, what is missing is completely analytical proof of the case $n=23$. So when Bushell wrote that $n=23$ is open, he meant that finding a completely analytical proof to be an open problem.

In the article linked to above, Fink summarizes Shapiro's inequality, and mentions several milestones of progress on it. Indeed, he calls the inequality "settled", which we may take to assume that the cases where it is true have all been proved, either analytically or by hinging upon computation.


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