注记1. 这种证明来自于一道课本习题：设http://data.artofproblemsolving.com/images/latex/0/8/8/0889b92c767302fe269a4b8df2aaaa30bfcb54ef.gif，则

http://data.artofproblemsolving.com/images/latex/4/3/7/4377faf2088746b0da530c538092ae81c313cb5e.gif

 这由 http://data.artofproblemsolving.com/images/latex/2/2/e/22ee809530cdbfcb3ded3b8b96a2057e2e00f41b.gif 即可证得。

所以

http://data.artofproblemsolving.com/images/latex/a/1/f/a1f511974308d1b81afc6ee967194b34d7a12e66.gif

    注记2：Nesbitt不等式的加强：

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=478373&p=2680410&hilit=Nesbitt's+Inequality#p2680410

    （1）.若 http://data.artofproblemsolving.com/images/latex/a/b/e/abe76d2c77a09951f2e6c54e369d6020064e63b1.gif ，   则 

http://data.artofproblemsolving.com/images/latex/d/8/e/d8e9e342241e4bc1045430c004617e37786c25b3.gif .

   它等价于Schur不等式

   http://data.artofproblemsolving.com/images/latex/0/6/f/06f0999d0b5f010ec26a072b37afbfabb5864ac6.gif

    （2 ）. 已知 http://data.artofproblemsolving.com/images/latex/2/e/c/2ecf9a885868b29bc73bb3ea3fdbabe7c9dedcab.gif则

 http://data.artofproblemsolving.com/images/latex/d/4/0/d4078c7afca965b0eb2eec865ce9f53fef98a91c.gif

     杨学枝《数学奥林匹克不等式研究》P260第73题，给出了（2）的证明：

http://s10/middle/547541cehc5f603541939&690of squares：平方和）证明" TITLE="不等式问题研究0：Nesbitt不等式的SOS(sum of squares：平方和）证明" />

     邹守文  Nesbitt不不等式的加强及应用《数学通讯》2012（4）P63给出了（2）的证明及应用。

    注记3：Sos - sum of squares方法详见

 http://www.artofproblemsolving.com/Forum/viewtopic.php?f=55&t=80127

    注记4：宋庆先生的证明：

(http://blog.sina.com.cn/s/blog_4c11310201011s6w.html)

 http://data.artofproblemsolving.com/images/latex/3/0/7/307543f7551f7fc2ca0008d8e74a33f27e61af10.gif

 http://data.artofproblemsolving.com/images/latex/a/f/b/afb6d4279bf8dd59c25d7fc7c95e5fc02595a729.gif

 http://data.artofproblemsolving.com/images/latex/a/4/4/a44415db9e0e3d9751dc4fd881a4e52b91cdb1ae.gif

 http://data.artofproblemsolving.com/images/latex/9/1/e/91e4429042ff7faa8bd3224606d7fb0609f3e339.gif

    宋庆博文中的注记：http://data.artofproblemsolving.com/images/latex/2/9/3/29330ac23972c3ef4ca3605987f3266218f36cc2.gif中有

http://data.artofproblemsolving.com/images/latex/0/3/4/03487c536861ad53c83c9a17d4ff8e16f0806592.gif  

而 

http://data.artofproblemsolving.com/images/latex/9/1/e/91e4429042ff7faa8bd3224606d7fb0609f3e339.gif

mudok刚刚提出不等式：已知http://data.artofproblemsolving.com/images/latex/9/c/a/9caa91157421e243281346b0bf7a82b5200e67e2.gif 是 http://data.artofproblemsolving.com/images/latex/2/9/3/29330ac23972c3ef4ca3605987f3266218f36cc2.gif 的三边长. 求证：

 http://data.artofproblemsolving.com/images/latex/a/b/9/ab97af9bf615d3fa5137342f665c36c8f6473c76.gif . 

http://data.artofproblemsolving.com/images/latex/b/1/9/b19bf3183c1043f2dc514ab8f35ff8e0097d7b17.gif

 再贴两个证明：（陈计《代数不等式》）

http://s5/middle/547541cehc5f55473fe04&690of squares：平方和）证明" TITLE="不等式问题研究0：Nesbitt不等式的SOS(sum of squares：平方和）证明" />

http://s8/middle/547541cehc5f55490ca77&690of squares：平方和）证明" TITLE="不等式问题研究0：Nesbitt不等式的SOS(sum of squares：平方和）证明" />