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Is this proposition established?

Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series $$p(x)=\sum_{n=0}^\infty a_nx^n,$$ $$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+1+\nu)}a_nx^{n+\nu}.$$ For any $a\in[0,1)$ and any sufficiently small $\delta>0$, there exists a certain $C>0$ such that $$\sup_{x,y\in[a,a+\delta]}|P(x)-P(y)|\geq C\sup_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}. $$

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 $$\sup_{x,y\in[a,a+\delta]}|P(x)-P(y)|\geq C\sup_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}. $$

Is this proposition established?

Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series $$p(x)=\sum_{n=0}^\infty a_nx^n,$$ $$P(x)=\sum_{n=0}^\infty \frac{1}{n+\nu}a_nx^{n+\nu}.$$ For any $a\in[0,1)$ and any sufficiently small $\delta>0$, there exists a certain $C>0$ such that $$\sup_{x,y\in[a,a+\delta]}|P(x)-P(y)|\geq C\sup_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}. $$

Is this proposition established?

Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series $$p(x)=\sum_{n=0}^\infty a_nx^n,$$ $$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+1+\nu)}a_nx^{n+\nu}.$$ For any $a\in[0,1)$ and any sufficiently small $\delta>0$, there exists a certain $C>0$ such that $$\sup_{x,y\in[a,a+\delta]}|P(x)-P(y)|\geq C\sup_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}. $$