\qquad若正实数 x、y、zx、y、zx、y、z 满足 xyz=1xyz=1xyz=1, 证明:
1x+y20+z11+1y+z20+x11+1z+x20+y11≤1.\frac1{x+y^{20}+z^{11}}+\frac1{y+z^{20}+x^{11}}+\frac1{z+x^{20}+y^{11}}\leq1.x+y20+z111+y+z20+x111+z+x20+y111≤1.
\qquad证明\qquad由柯西不等式得 1a+b20+c11≤a13+b−6+c3(a7+b7+c7)2(a、b、c∈R+).\frac1{a+b^{20}+c^{11}}\leq\frac{a^{13}+b^{-6}+c^3}{(a^7+b^7+c^7)^2} (a、b、c\in\R^+).a+b20+c111≤(a7+b7+c7)2a13+b−6+c3(a、b、c∈R+).
\qquad当 (a,b,c)=(x,y,z),(y,z,x),(z,x,y)(a,b,c)=(x,y,z),(y,z,x),(z,x,y)(a,b,c)=(x,y,z),(y,z,x),(z,x,y) 时,求和得
\qquad1x+y20+z11+1y+z20+x11+1z+x20+y11\frac1{x+y^{20}+z^{11}}+\frac1{y+z^{20}+x^{11}}+\frac1{z+x^{20}+y^{11}}x+y20+z111+y+z20+x111+z+x20+y111
\qquad≤x13+y−6+z3(x7+y7+z7)2+y13+z−6+x3(x7+y7+z7)2+z13+x−6+y3(x7+y7+z7)2.\leq\frac{x^{13}+y^{-6}+z^3}{(x^7+y^7+z^7)^2}+\frac{y^{13}+z^{-6}+x^3}{(x^7+y^7+z^7)^2}+\frac{z^{13}+x^{-6}+y^3}{(x^7+y^7+z^7)^2}.≤(x7+y7+z7)2x13+y−6+z3+(x7+y7+z7)2y13+z−6+x3+(x7+y7+z7)2z13+x−6+y3.
\qquad因此,只需证
\qquadx13+y13+z13+x−6+y−6+z−6+x3+y3+z3x^{13}+y^{13}+z^{13}+x^{-6}+y^{-6}+z^{-6}+x^3+y^3+z^3x13+y13+z13+x−6+y−6+z−6+x3+y3+z3
\qquad≤x14+y14+z14+2(x7y7+y7z7+z7x7).\leq x^{14}+y^{14}+z^{14}+2(x^7y^7+y^7z^7+z^7x^7).≤x14+y14+z14+2(x7y7+y7z7+z7x7).
\qquad因为 xyz=1xyz=1xyz=1, 所以,
\qquad x13+y13+z13=∑x1313y13z13,x^{13}+y^{13}+z^{13}=\sum x^{13\frac13}y^{\frac13}z^{\frac13},x13+y13+z13=∑x1331y31z31,
\qquadx−6+y−6+z−6=∑x623y623z23x^{-6}+y^{-6}+z^{-6}=\sum x^{6\frac23}y^{6\frac23}z^{\frac23}x−6+y−6+z−6=∑x632y632z32
\qquadx3+y3+z3=∑x623y323z323.x^3+y^3+z^3=\sum x^{6\frac23}y^{3\frac23}z^{3\frac23}.x3+y3+z3=∑x632y332z332.
\qquad又 (1313,13,13)<(14,0,0),(623,623,23)<(7,7,0),(623,323,323)<(7,7,0)(13\frac13,\frac13,\frac13)<(14,0,0),(6\frac23,6\frac23,\frac23)<(7,7,0),(6\frac23,3\frac23,3\frac23)<(7,7,0)(1331,31,31)<(14,0,0),(632,632,32)<(7,7,0),(632,332,332)<(7,7,0)
\qquad由 Muirhead\rm {Muirhead}Muirhead 不等式得
\qquad∑x1313y13z13≤∑x14y0z0\sum x^{13\frac13}y^{\frac13}z^{\frac13}\leq\sum x^{14}y^0z^0∑x1331y31z31≤∑x14y0z0
\qquad∑x623y623z23≤∑x7y7z0\sum x^{6\frac23}y^{6\frac23}z^{\frac23}\leq\sum x^7y^7z^0∑x632y632z32≤∑x7y7z0
\qquad∑x623y323z323≤∑x7y7z0\sum x^{6\frac23}y^{3\frac23}z^{3\frac23}\leq\sum x^7y^7z^0∑x632y332z332≤∑x7y7z0
[注] Muirhead\rm {Muirhead}Muirhead 不等式见 1965 年科学出版社出版的 G.H⋅\rm{G.H}\cdotG.H⋅哈代,J.E⋅\rm{J.E}\cdotJ.E⋅李特伍德,G⋅\rm {G}\cdotG⋅波利亚著,越民义译《不等式》一书第 46 页.