I am reading the article which are related to nonparametric estimation of the fragmentation kernel based on a partial differential equation. I have some difficulties in understanding some following lines.
First, let me recall the definition of Holder class
The function $h$ is of class $C^{\beta}$ on $[0;1]$, for some $\beta >3$: The function $h$ is $\left[ \beta\right]$ times differentiable (where $\left[\beta\right]$ is the largest integer smaller than $\beta$) and the derivative of order $\left[\beta\right]$ is $\beta - \left[\beta \right]$ Holder continuous.
Next, moreover, we assume that there exists a positive integer $v_0 \ge 2$ such that for all $k \in {0,....,v_0}$ we have $h^{k}(0)=0$.
In the article, they claimed that
Using a Taylor expansion, it implies that for any $t \in (0;1)$ we have $$\int_0^t h(x) d x \leq C \int_0^t x^{\left(v_0+1\right) \wedge[\beta]} d x$$
I don't know why they can obtain this result. Here is my attempt: I intend to use Taylor's formula. That means for $t\in(0;1)$, there exists indeed $\theta \in (0;1)$ such that $$0 \leq \frac{h(t)}{t}=\sum_{k=v_0+1}^{[\beta]-1} \frac{1}{k !} h^{(k)}(0) t^{k-1}+\frac{h^{[\beta]}(\theta t)}{[\beta] !} t^{[\beta]-1}$$ Next, I am going to take the integral but maybe it is not enough to achieve the result. Thank you for reading my post. Any help is appreciated. On the other hand, for your reference, here is the article which I mentioned in the previous part. Besides, the mentioned statement is on page 32.