Consider
$$ f(w,L) = \int_1^w \int_0^{2 L \pi} \frac{ \ln(\frac{sin(x) + sin(vx)}{2} + \frac{5}{4})}{L(w - 1)} dx dv $$
For real $w > 1 $ and integer $ L > 1$
Conjecture :
$$ \lim_{L \to \infty} f(w + 1, L) - f(w,L) = 0. $$
How to decide if this is true ?
Perhaps differentiation under the integral sign is the best method ?
——
To see where “ this is coming from “ ,
Notice
$$\int_0^{2 \pi} \ln(sin(x) + \frac{5}{4}) dx = 0 $$
And look at these :
Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
Why is $\inf g \sup g = \frac{9}{16} $?
—-