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It's a follow up of Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$ :

Problem :

Let $x,y,z,k>0$ find the "best" value of $k$ such that :

$$\frac{x^4}{kx^3+y^3} + \frac{y^4}{ky^3 + z^3} + \frac{z^4}{kz^3 + x^3} \ge \frac{x+y+z}{k+1}.$$

The inequality has a local minimum around $(x,y,z)=\left(\frac{43}{66},1,\frac{121}{84}\right)$ in the case where $k=\frac{133}{81}$

Some tought :

We have a first fact :

$$f(x):=\frac1{150607+91722x^6}.$$ The function is convex on $x\in[1.03,\infty)$.

Proof :

$$f''(x)=\frac{550332(642054x^6-753035)}{150607+91722x^6}$$

Then we minimize using Jensen's inequality. We have the new problem for $x,y,z\geq 0$ and $z\geq y \ge x$, $\frac{z}y,\frac{y}x\in[1.031,\infty)$:

$$g(x,y,z)=\left(x^2+0.000001\right)f\left(\frac{0.000001\frac{z}y+xy}{x^2+0.000001}\right)+0.999999f\left(\frac{z}y\right)+\frac{z^8}{150607z^6+91722x^{6}}-\frac{x^2+y^2+z^2}{150607+91722}$$

Then (and it includes the hard case described by user RiverLi) :

$$g(x,1,(1+x^8)(1+y))\geq0$$

Where all the coefficients are positive.

We can use this method as we wish, but it doesn't really give the value of $k$ .

Edit :

Using basic algebra and derivative we have the function $x,y,z,p,u,v,k,w,n>0$:

$$f(x)=\frac{px^4}{x^3+1}+u\cdot\frac{y^3}{y^3+1}+v\frac{\frac{z^3}y}{z^3+1}-\frac{1+pkx+\frac{w}y}{n+1}$$

$xyz=\text{constant}$

For the minimum see here.

Motivation :

In the post linked, tthnew gives a polynomial wich gives the value of $k\simeq\frac{150607}{91722}$ . So I think like RiverLi: there's no closed form using radicals. Now, there is here the Galois's theory even if to all confess it's far beyond my level .Anyway the goal idea is : we can find(introducing a $\varepsilon$ in the new problem/inequality) for an arbitrary large number of digits (1.6,1.64,1.649 and so on ) the value of $k$.That is a problem of limit when $\varepsilon\to zero$.Another argument you can find in the edit is :using derivatives and a cheated constraint we can calculate all the extrema wich is good news, but from my point of view leads nowhere and needs another argument. So, my motivation is : using classical tools (derivative, Jensen,...) rather than "heavy" theory (Galois,elliptic..) it would be a "tour de force" if we find the value of $k$ like this.

Edit : I finally found a way using Lagrange Reversion theorem apply to tthnew's equation .

Question :

How to find the "best" constant $k$ ,here I mean by "best" wich cannot be replaced by a stronger value .

And is there any closed form for $k$ ?

Barackouda
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  • Suggestion: (1) You should use punctuation marks. For example, "The function is convex for $x\in[1.03,\infty)$." (a period at the end) (2)"Where all the coefficients are positives." The "Where" should be "where" because it is not a new sentence. (3) at the end of each equation, add a period. for example, $$f\left(x\right)=\frac{1}{150607+91722x^{6}}.$$ – River Li Jun 28 '22 at 13:55
  • You can see e.g. @Carl Schildkraut's answers in MSE, e.g. https://math.stackexchange.com/questions/4341574/lim-n-to-infty-frac1n-sum-k-1nfk-where-fn-is-the-largest-p, and https://math.stackexchange.com/questions/4456286/prove-that-fraca-sqrta2b2-fracb-sqrtb2c2-fracc-sqrtc2/. – River Li Jun 28 '22 at 13:57
  • (4) Remark for (3), there is sometimes no punctuation mark or add a comma at the end of an equation. For example, $$g(x,1,(1+x^8) (1+y))\geq0$$ where all the coefficients are positives. – River Li Jun 28 '22 at 14:04
  • By the way, I did not downvote you although I commented (criticized) on your writing. – River Li Jun 29 '22 at 06:00
  • @RiverLi Thanks for the comment . – Barackouda Jun 29 '22 at 07:57
  • @RiverLi Can you explain why sometimes people downvote something true ? – Barackouda Jun 29 '22 at 08:08
  • @Invisible I'm not against criticism . I have a big lack a rigor and that's why you come to me . I note all your comment but I haven't the time yet to correct . On the other hand I don't put myself in somes essential question (for the reader) and now I understand your light . Thanks you very much . – Barackouda Jun 29 '22 at 08:54
  • @ErikSatie Did you see my last comment. It should be "$g(x,1,(1+x^8) (1+y))\geq0$ (without a period here) where blablabla". Also, as I pointed out before, some parentheses are not necessary, for example, $\frac{\left(x^{2}+y^{2}+z^{2}\right)}{150607+91722}$, $\frac{\left(1+pkx+\frac{w}{y}\right)}{n+1}$. – River Li Jun 29 '22 at 10:04
  • You don't have to prove $f$ is convex. It is more important to write *why you introduce or how you came up with the function $f$. When I look at it, without reading your answers in the link (and it's a bit of a problem on phone because loading of the enormous amount of MathJax takes too much time). We write functions the way we do exactly to avoid writing parentheses. And the numbers are so large that one forgets what you wrote half a line prior. Keep it short and sweet. – Matcha Latte Jun 29 '22 at 19:32
  • @ErikSatie tthnew’s equation says “Sorry, the file you have requested does not exist.” – Тyma Gaidash Jan 17 '23 at 13:00

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