Consider this question here :
Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
Call that conjecture about $\frac{5}{4} $ conjecture $1$.
Let $g(n) = \prod_{i=0}^n (\sin^2(n) + \frac{9}{16}) ) $
Conjecture $3$ :
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Conjecture $2$ is :
$$ \sup g(n) \space \inf g(n) = \frac{9}{16} $$
And this follows from conjecture $1$ or vice versa.
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It feels like this second conjecture could somehow follow from the first conjecture since
$$-(\cos(n) + \frac{5}{4})(\cos(n) - \frac{5}{4}) = - \cos^2(n) + \frac{25}{16} = \sin^2(n) + \frac{9}{16} $$
This question is about the connection ( conjecture $3$).
If you can prove conjecture $1$ or $2$ post it in the other thread.
Btw $ \int_0^{2 \pi} \ln(\sin^2(x) + \frac{9}{16}) dx = 0 $ indeed as you probably already knew or guessed.
