Is this proposition established?


Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series
$$p(x)=\sum\_{n=0}^\infty a\_nx^n,$$
$$P(x)=\sum\_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+1+\nu)}a\_nx^{n+\nu}.$$
Suppose that $p'(x),P'(x)$ don't exist. For any $a\in[0,1)$ and any sufficiently small $\delta>0$, there exists a certain $C>0$ such that
$$\sup\_{x,y\in[a,a+\delta]}|P(x)-P(y)|\geq C\sup\_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}. $$



A proposition about power series
If $P'(a)=0\neq p'(a)$, then there is no such constant. Indeed, in this situation, we have for sufficiently small $\delta$,
\begin{align\*}
\sup\_{x,y\in[a,a+\delta]}|P(x)-P(y)|&\ \ll\_a\ \delta^2\\
\sup\_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}&\ \gg\_a\ \delta^{1+\nu}.
\end{align\*}
These bounds follow readily from the Taylor series expansion of $P(x)$ and $p(x)$ around $a$. In particular, the ratio of the left hand sides tends to zero under $\delta\to 0+$, hence it is not bounded away from zero.


