Inequality $3\sin^{-1}\left(\frac{\pi^{1/3}}{2^{2/3}}\right)\leq a+b+c$ with some conditions

Question

I have a new problem create by myself :

Let $0<a,b,c<\frac{\pi}{2}$ such that $\sin(a)\sin(b)\sin(c)=\frac{\pi}{4}$ then we have : $$3\sin^{-1}\left(\frac{(\pi)^{\frac{1}{3}}}{2^{\frac{2}{3}}}\right)\leq a+b+c$$

I have tried to use the logarithm function : $$\ln(\sin(a))+\ln(\sin(b))+\ln(\sin(c))=\ln(\frac{\pi}{4})$$

And use the concavity of $\ln(\sin(x))$ we get :

$$\ln(\sin(a))+\ln(\sin(b))+\ln(\sin(c))\leq 3\ln\left(\sin\left(\frac{a+b+c}{3}\right)\right)$$

Now it's easy to conclude .

My question have you an alternative way ?

Thanks a lot !

Answer 1

We can use the Tangent Line method here: $$\sum_{cyc}\left(a-\arcsin\sqrt[3]{\frac{\pi}{4}}\right)=\sum_{cyc}\left(a-\arcsin\sqrt[3]{\frac{\pi}{4}}-\frac{\ln\sin{a}-\ln\sqrt[3]{\frac{\pi}{4}}}{\sqrt{\sqrt[3]{\frac{16}{\pi^2}}-1}}\right)\geq0$$ because $$a-\arcsin\sqrt[3]{\frac{\pi}{4}}-\frac{\ln\sin{a}-\ln\sqrt[3]{\frac{\pi}{4}}}{\sqrt{\sqrt[3]{\frac{16}{\pi^2}}-1}}\geq0$$ for any $0<a<\frac{\pi}{2}.$

Indeed, let $f(a)=a-\arcsin\sqrt[3]{\frac{\pi}{4}}-\frac{\ln\sin{a}-\ln\sqrt[3]{\frac{\pi}{4}}}{\sqrt{\sqrt[3]{\frac{16}{\pi^2}}-1}}.$

Thus, easy to see that $$f\left(\arcsin\sqrt[3]{\frac{\pi}{4}}\right)=0,$$ $$f'\left(\arcsin\sqrt[3]{\frac{\pi}{4}}\right)=0$$ and for all $0<a<\frac{\pi}{2}$ $$f''\left(a\right)>0.$$