Assume that $p\in(1,2]$, $a,b\ge 1$, $p\in[1,2]$ and$b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{p}}\right)^{p/2}\geq \left(\frac{a+\left(a b-\sqrt{a^2-1} \sqrt{b^2-1}\right) \cos\frac{\pi }{p}}{b+\cos\frac{\pi }{p}}\right)^{p/2}+\frac{\cos\frac{p t}{2} \sin\frac{\pi }{p}}{b+\cos\frac{\pi }{p}}$$$$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{p}}\right)^{p/2}\geq \left(\frac{a+\left(a b-\sqrt{a^2-1} \sqrt{b^2-1}\right) \cos\frac{\pi }{p}}{b+\cos\frac{\pi }{p}}\right)^{p/2}+\frac{\cos\frac{p t}{2} \sin\frac{\pi }{p}}{b+\cos\frac{\pi }{p}},$$
