# The parameter regularity of power sum

Let $$f(x,s)=\sum_{n=0}^\infty a_n(s)x^n$$ where $$|a_n(s)|\le1$$ is a bounded function theory. Suppose for every $$|x|<1$$, $$f(x,s)$$ is Holder-$$\alpha$$ for $$s$$-variable, i.e. $$|f(x,s_1)-f(x,s_2)|\le C|s_1-s_2|^\alpha$$. Then can we have $$a_n(s)$$ is Holder-$$\alpha$$ for every $$n$$?

Equivalently, can we have, for every $$z\in\mathbb C$$, $$|z|<1$$, $$f(z,s)$$ is of Holder-$$\alpha$$ for $$s$$?

• You mean for $x\in(-1,1)$? Also, is it OK to assume that the estimate is uniform for $|x|\le r$ when $r<1$, or you want it exactly as written? – fedja Jun 27 '19 at 4:19
• @fedja Yes the radius does not really matters, all I need is the local convergence – yaoliding Jun 27 '19 at 4:20
• Anyway, the answer is "No". All you can get is that the function in the complex domain is $\beta$-Holder with any $\beta<\alpha$ in a sufficiently small disk depending of $\beta$. – fedja Jun 27 '19 at 4:42
• @fedja But what is the example – yaoliding Jun 27 '19 at 4:43
• Consider $f(z)=\sum_{m\ge 1}b_me^{-m}e^{-imz}\min(1,e^m s)$ with $b_m=m^{-2}$, say and $s\in[0,1]$. It is bounded in the strip $|\Im z|\le 1$, Lipschitz in $s$ uniformly on the real line but only $\frac 12$-Holder in $s$ (at $s=0$) on $\mathbb R+\frac i2$. – fedja Jun 27 '19 at 4:57