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Hi it's related to the following conjecture An inequality for polynomials with positives coefficients :

We have the first conjecture :

Let $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+y^2}{x+y}\Big)(f(x)+f(y))\geq 2(xf(x)+yf(y))f\Big(\frac{x+y}{2}\Big)$$ Where :$$f(x)=\sum_{i=0}^{n}a_ix^i$$ And $a_0 \ge a_1 \ge ... \ge a_n>0$ (Thanks to p4sch)

Second conjecture :

Let $x_i>0$ be $n$ real numbers then we have : $$(\sum_{i=1}^{n}x_i)f\Big(\frac{\sum_{i=1}^{n}x_i^2}{\sum_{i=1}^{n}x_i}\Big)(\sum_{i=1}^{n}f(x_i))\geq n(\sum_{i=1}^{n}x_if(x_i))f\Big(\frac{\sum_{i=1}^{n}x_i}{n}\Big)$$ Where :$$f(x)=\sum_{i=0}^{n}a_ix^i$$ And $a_0 \ge a_1 \ge ... \ge a_n>0$

It seems to be true in the case where $f(x)=\tan(x)$ see A sharp inequality for tangent (refinement of Jensen's inequality)

I have tried the approach of River Li in the case $n=3$ without success . I think it's really hard and interesting since at the beginning I have put together two applications of Jensen's inequality .

Putting $f(x)=e^x$ gives a nice refinement of Am-Gm (as applications).

If it's true I would like just one approach to prove it .

Any helps is greatly appreciated .

Thanks a lot for all your contributions .

Ps:Is it a good idea to put it on maths overflow ?

Reference :

https://en.wikipedia.org/wiki/Stable_polynomial

Barackouda
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    Maybe ti's better to wait a moment for answers on Math Stack Exchange before attempting another site. – Jean Marie May 16 '20 at 18:08
  • What has your conjecture to do with stable polynomials? – Martin R Feb 26 '24 at 09:43
  • @DesmosTutu it went unanswered for a long time. Do you think it can be answered now? I will take a look in detain once I have more time... – D S Feb 26 '24 at 13:04
  • @DS it seems we have : $$g(x)=\frac{f\left(\frac{x^2+y^2}{x+y}\right)}{f(x/2+y/2)}\geq \frac{g(x)-h(x)}{\ln(g(x))-\ln(h(x))}\geq h(x)=\frac{2(xf(x)+yf(y))}{(x+y)(f(x)+f(y))},x\neq y ,x>0,y>0$$ and the constraint on the polynomial above – Barackouda Feb 26 '24 at 19:56
  • @DS Can you make a proof with my partial answer ? – Barackouda Feb 27 '24 at 19:53
  • @DesmosTutu I have exams rn, and tmrw is a difficult one. Your question seems difficult, though I don't think I understand what we have to prove. The first conjecture has already been disproved in the linked post, so you've added a new condition in it $a_0\ge a_1\ge a_2\ldots$, am I right? And then you're asking about the second conjecture too (a stronger one)? – D S Feb 27 '24 at 20:29
  • @DS OK .the first conjecture implies the second with LCF theorem by Vasile Cirtoaje .I come back if I have new things . – Barackouda Feb 28 '24 at 07:39

1 Answers1

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Partial answer

I provide a partial proof for :

$$f(x)=1+x+x^2$$

$$Difference=(x-y)^4×\frac{positivepolynomial}{2(x+y)}$$

Now we can enlarge the proof using homogeneity or $y=aY,x=aX$

It's highly generalizable

A Conjecture:

Let the D be the difference between the two side above then :

There exists $C_n>0$ constants and $k,m$ positive integers and $n>2$ Then:

$$D=(x-y)^m\frac{positivepolynomial}{C_n(x+y)^k}$$

It seems $m=4$ if so we can start with the case where $a_i=1$

Barackouda
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