hello, it is equivalent with $(x-y)(\cos(y)-\cos(x))+(x-z)(\cos(z)-\cos(x))+(y-z)(\cos(z)-\cos(y))\geq 0 $ if $x\geq y\geq z$ Sonnhard.

joybangla wrote: $x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that: \[ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} \] It's exactly Chebyshov for $(x,y,z)$ and $(\cos x, \cos y, \cos z)$.