Question Number 219988 by Nicholas666 last updated on 04/May/25
![let s>1 be a real number. for all continues function f:[0,1]→R such that ∫_( 0) ^( 1) f(x)=0, determind of the exist a positive constant K(s) statisfying: (∫_0 ^( 1) f(x)∙Li_s (x)dx)^2 ≥K(s)∫_( 0) ^( 1) (f(x))^2 ∙Li_(2s−1 ) where Li_s (x)=Σ_(k=1) ^∞ (x^k /k^s ) is the polylogarithm function. if such a constants exists, find the optimal value of K(s).](https://www.tinkutara.com/question/Q219988.png)
$$ \\ $$$$\:\:\:\:\mathrm{let}\:{s}>\mathrm{1}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}.\:\mathrm{for}\:\mathrm{all}\:\mathrm{continues}\:\mathrm{function}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R} \\ $$$$\:\:\:\mathrm{such}\:\mathrm{that}\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} {f}\left({x}\right)=\mathrm{0},\:\mathrm{determind}\:\mathrm{of}\:\mathrm{the}\:\mathrm{exist}\:\mathrm{a} \\ $$$$\:\:\:\:\:\mathrm{positive}\:\mathrm{constant}\:{K}\left({s}\right)\:\mathrm{statisfying}: \\ $$$$\:\:\:\left(\int_{\mathrm{0}} ^{\:\mathrm{1}} {f}\left({x}\right)\centerdot\mathrm{Li}_{{s}} \left({x}\right){dx}\right)^{\mathrm{2}} \geqslant{K}\left({s}\right)\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \left({f}\left({x}\right)\right)^{\mathrm{2}} \centerdot\mathrm{Li}_{\mathrm{2}{s}−\mathrm{1}\:} \\ $$$$\:\:\:\mathrm{where}\:\mathrm{Li}_{{s}} \left({x}\right)=\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{k}} }{{k}^{{s}} }\:\mathrm{is}\:\mathrm{the}\:\mathrm{polylogarithm}\:\mathrm{function}.\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\mathrm{if}\:\mathrm{such}\:\mathrm{a}\:\mathrm{constants}\:\mathrm{exists},\:\mathrm{find}\:\mathrm{the}\:\mathrm{optimal}\:\mathrm{value}\:\mathrm{of}\:{K}\left({s}\right).\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Commented by Mamadi last updated on 04/May/25
$${thank}\:{you} \\ $$
Answered by MrGaster last updated on 04/May/25
$$\mathrm{The}\:\mathrm{following}\:\mathrm{results}\:\mathrm{can}\:\mathrm{beb} \\ $$$$\mathrm{otained}\:\mathrm{by}\:\mathrm{computern} \\ $$$$\mathrm{explosio}\:\mathrm{calculation}: \\ $$$$\mathrm{1}.{K}\left({s}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left(\mathrm{Li}_{{s}} \left({x}\right)\right)^{\mathrm{2}} }{\mathrm{Li}_{\mathrm{2}{s}−\mathrm{1}} \left({x}\right)}{dx}−\frac{\left(\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{Li}_{{s}} \left({x}\right){dx}\right)^{\mathrm{2}} }{\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{Li}_{\mathrm{2}{s}−\mathrm{1}} \left({x}\right){dx}} \\ $$$$\mathrm{2}.{K}\left({s}\right)=\frac{\mathrm{1}}{\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} \left({k}+\mathrm{1}\right)}−\left(\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{k}^{{s}} \left({k}+\mathrm{1}\right)}\right)^{\mathrm{2}} } \\ $$$$\mathrm{3}.{K}\left({s}\right)=\frac{\mathrm{1}}{\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left(\mathrm{Li}_{{s}} \left({x}\right)−\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{Li}_{{s}} \left({t}\right){dt}\right)^{\mathrm{2}} }{\mathrm{Li}_{\mathrm{2}{s}−\mathrm{1}} \left({x}\right)}{dx}} \\ $$$$\mathrm{4}.{K}\left({s}\right)=\frac{\mathrm{1}}{\zeta\left(\mathrm{2}{s}\right)} \\ $$$$ \\ $$$$ \\ $$
Commented by MrGaster last updated on 04/May/25
Personally, this topic belongs to the level of graduate students or above, because it requires a lot of knowledge of functional analysis.And the expression itself is not elementary.It is not appropriate to put it here, at least on a platform that is completely related to high-number content.
Commented by Nicholas666 last updated on 04/May/25
$${yes}\:{sorry}\:{sir}. \\ $$$${I}\:{took}\:{this}\:{problem}\:{from}\:{my}\:{book}. \\ $$$${thank}\:{you}\:{for}\:{you} \\ $$