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

  • $\begingroup$ What is $C$ supposed to depend on? As stated, $C$ may depend on $(a_n)$, $a$ and $\delta$, in which case such $C$ obviously exists. $\endgroup$ – Mateusz Kwaśnicki Feb 14 at 15:11
  • 1
    $\begingroup$ It does not makes sense to assume that $p'(x)$ does not exist. If $p(x)$ converges in $(0,1)$, then the function defined by it is infinitely many times differentiable in $(0,1)$. In fact $p(x)$ extends holomorphically to the unit disk $\{z\in\mathbb{C}:|z|<1\}$. This is one of the basic theorems in complex analysis. See en.wikipedia.org/wiki/… See also my last comment under my response. $\endgroup$ – GH from MO Feb 17 at 23:18

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.

  • $\begingroup$ What if P'(x) and p'(x) do not exist? $\endgroup$ – Burnie Feb 14 at 15:07
  • $\begingroup$ @Burnie: They always exist for $x \ne 0$, do they not? $\endgroup$ – Mateusz Kwaśnicki Feb 14 at 15:12
  • $\begingroup$ the convergence of Σan does not indicate a convergence of Σn*an. @Mateusz Kwaśnicki $\endgroup$ – Burnie Feb 14 at 15:14
  • $\begingroup$ @Burnie: Any power series is differentiable in the interior of its set of convergence. Hence $P'(a)$ and $p'(a)$ exist. More generally, if $u_n$ are holomorphic functions on an open set $M\subset\mathbb{C}$, and the function series $\sum u_n$ converges locally uniformly on $M$, then the series defines a holomorphic function on $M$, and its derivative equals $\sum u_n'$ on $M$, which itself converges locally uniformly on $M$. $\endgroup$ – GH from MO Feb 14 at 19:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.