I'm interested by the following problem :

Let $x,y>0$ such that $x<\dfrac{\pi}{4}$, $y<\dfrac{\pi}{4}$ then we have : $$\ln(\sin\Bigg(\frac{\frac{x}{\tan(x)}+\frac{y}{\tan(y)}}{\frac{1}{\tan(x)}+\frac{1}{\tan(y)}}\Bigg))+\ln(\sin \bigg(\frac{x+y}{2} \bigg))\leq \ln(\sin(x))+\ln(\sin(y))$$

Since $\ln(\sin(x))$ is concave with the condition I would apply the Jensen's inequality but it's too weak so I'm a bit lost . If you have a hint it would be nice . Thanks a lot for your time .


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