Summability of ratios of moments a weight Recently, I encounter the following problem:
Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e.,
$$m_k=\int_0^1t^kw(t)dt.$$
Under what condition can we have
$$\sum_{k=0}^\infty \frac{m_{2k}−2m_{2k+1}+m_{2k+2}}{m_{2k+1}}< \infty ?$$
If necessary and sufficient condition is not known, what kind of sufficient conditions and what kind of necessary conditions are known?
I know that weight functions like $c_\alpha \cdot (1−t)^\alpha$ with $\alpha>−1$ are such examples.
Thanks in advance for any reference. 
 A: Let first state an equivalent condition. Consider the function $\phi:[0,1]\to[0,+\infty]$ given by
$$\phi(t):=\sum_{k=0}^\infty \frac{t^{2k}}{m_{2k+1}}\, .$$
By the Beppo Levi theorem,
$$ \sum_{k=0}^\infty \frac{m_{2k}-2m_{2k+1}+m_{2k+2}}{m_{2k+1}}=
\sum_{k=0}^\infty  \frac{1}{m_{2k+1}}\int_I t^{2k}(1-t)^2w(t) dt $$
$$=\int_I\phi(t)(1-t)^2w(t) dt\, .$$ 
So the condition can be restated as $\phi(t)(1-t)^2$ being integrable w.r.to $w(t)dt$, which turns out to be  a regularity condition for  $w(t)$ at $t=1$ .  
Here is  a sufficient condition in terms of $w$. Assume, for $\alpha>-1$ and $\alpha-\beta< 1$
$$ c(1-t)^{\alpha }\le w(t)\le C(1-t)^{\beta }\; ,  $$  locally at $t=1$, for some positive constants $c$ and $C$.
Then by standard asymptotics $m_k\ge C_1 B(\alpha+1,k+1))\ge C_2 k^{-\alpha-1}$ , and comparing $m_{2k+1}^{-1}$ with the coefficients of a binomial series one finds  $ \phi(t)=O((1-t)^{-\alpha-2})$ for $t\to 1$, so $(1-t)^2\phi(t)w(t)\le C_3(1-t)^{\beta-\alpha}$ on $I$, which is integrable.
