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](http://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0983563-0/S0025-5718-1989-0983563-0.pdf).


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](http://downloads.hindawi.com/journals/jia/2002/509463.pdf) 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](http://en.wikipedia.org/wiki/Shapiro_inequality) for whatever it is worth.



Shapiro inequality for $n=23$
There is no incongruity between the two papers that have been cited in the question. 


According to [A.M. Fink](http://books.google.de/books?id=xtqZ7_BfKJAC&lpg=PA247&ots=TJZeH60U0O&dq=shapiro%2520inequality&pg=PA247), 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.


