Let $a_n $ be a nondecreasing sequence of positive reals. Let $\sum a_n $ be divergant. Let $f(x)$ be real-analytic for $Re(x) > -1$. Also $f(0) = 0,f(1)=1$ and $f(x)$ is strictly increasing for $x>0$. Identical properties for $a(x)$ where $a(n) = a_n$.
If $ lim_{x->1^-}$ $\int_0^{\infty} \frac{a(y)}{f(xy - y+1)} dy = L $ exists with a finite $L$ and
If $ lim_{x->1^-}$ $\sum_{y=0}^{\infty} \frac{a(y)}{f(xy - y+1)} = L_1 $ exists with a finite $L_1$,
Does that imply
$(L_1 - L)^2 < (3 a(3) f(3) )^3 $ ? [1]
Also does the existance of $L$ resp $L_1$ imply the existance of the other ?
Does there always exist An $f(x)$ ( for every $a_n , a(x) $) such that [1] holds ?
These questions would be trivial if there was no divergeance ofcourse , but now I do not know how to solve these.