# $x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$

This inequality has been verified by Mathematica. $\frac12$ is not the best bound. I try to do AM-GM for this one but not yet success. The condition $x+y^2+z^3$ is very weird.

• When $x=0.534,y=0.5$ and $z=0.6$ (exact decimal values) one has $x^2y+y^2z+z^2x=0.484818$, so 0.5 is close to the best constant. – Ewan Delanoy May 21 '16 at 18:19
• Mathematica gives the maximum value of $x^2y+y^2z+z^2x < \frac12$ as constrained to be $0.48562209920309984177$ for $x=0.55811253$, $y=0.49841201$, $z=0.57837131$. – Steve Kass May 26 '16 at 3:23
• I think, this method will be interesting too. – Yuri Negometyanov Apr 6 '17 at 11:25
• A nice solution is also given by anhduy98@AoPS here: artofproblemsolving.com/community/c6h1238490 – River Li Nov 6 '20 at 2:47

The standard way of solving the problem on a conditional extremum is the method of Lagrange multipliers, which reduces it to a system of equations.

The greatest value of function $$f(x,y,z,\lambda) = x^2y+y^2z+z^2x+\lambda(x+y^2+z^3-1)$$ on the interval $$x,y,z\in[0,1]$$ is reached or at its edges, or in the inner stationary point.

$$\color{brown}{\textbf{Inner stationary points.}}$$

The inner stationary points has zero partial derivatives $$\begin{cases} f'_\lambda = x + y^2 + z^3 - 1 = 0\\ f'_x = z^2 + 2xy + \lambda = 0\\ f'_y = x^2 + 2yz + 2\lambda y = 0\\ f'_z = y^2 + 2zx + 3\lambda z^2 = 0. \end{cases}$$ After the excluding of parameter $$\lambda$$ get the system $$\begin{cases} x + y^2 + z^3 - 1 = 0\\ x^2 + 2yz = 2y(z^2 + 2xy)\\ y^2 + 2zx = 3z^2(z^2 + 2xy), \end{cases}$$ or $$\begin{cases} x + y^2 + z^3 - 1 = 0\\ (1-3y^2-z^3)^2-4y^4+2yz(1-z)=0\\ 2z(1-3yz)(1-y^2-z^3)+y^2-3z^4=0. \end{cases}$$

Using of Groebner basis allows to get the positive solutions $$\genfrac{[}{.}{0}{0}{x\approx 0.16367,\quad y\approx 0.761982,\quad z\approx 0.634724,\quad f\approx 0.454882} {x\approx 0.558113,\quad y\approx 0.498412,\quad z\approx 0.578371,\quad f\approx 0.485622}.$$ Alternative way is shown below.

$$\color{brown}{\textbf{The edges.}}$$

The edges of the field are achieved when $$x = 0$$, $$y = 0$$ or $$z = 0$$.

Substitution $$x = 0$$ in the expressions for the partial derivatives leads to the system $$\begin{cases} x=0\\ y^2+z^3 = 1\\ 2yz+2\lambda y = 0\\ y^2+3\lambda z^2 = 0 \end{cases}$$ with solutions $$\genfrac{[}{.}{0}{} {x=0,\quad y=\sqrt{0.75},\quad z=\sqrt[3]{0.25}\approx 0.629991,\quad f\approx 0.47247} {x=0,\quad y=0,\quad z=1,\quad f=0}$$

Substitution $$y = 0$$ in the expressions for the partial derivative leads to the system $$\begin{cases} y=0\\ x+z^3=1\\ z^2+\lambda = 0\\ 2zx+3\lambda z^2 = 0 \end{cases}$$ with solution $$x=0.6,\quad y=0,\quad z=\sqrt[3]{0.4}\approx 0.736806,\quad f\approx 0.32573.$$

Substitution $$z = 0$$ in the expressions for the partial derivative leads to the system $$\begin{cases} z=0\\ x+y^2=1\\ 2xy+\lambda = 0\\ x^2+2\lambda y = 0 \end{cases}$$ with solutions $$\genfrac{[}{.}{0}{} {x=0.8,\quad y=\sqrt{0.2}\approx 0.447214,\quad z=0,\quad f\approx 0.286217} {x=0,\quad y=0,\quad z=1,\quad f=0}$$

The values of function at the vertices of the area (unit parallelepiped) equal to zero.

So the greatest value approximately equals to $$0.485622$$. Given accuracy of calculations provides the inequality $$\boxed{x^2y+y^2z+z^2x<1/2.}$$

$$\color{brown}{\textbf{System resolving, alternative way.}}$$

Set condition allows us to reduce the problem to finding unconditional extremes of $$f(y,z)=(1-y^2-z^3)^2y+y^2z+z^2(1-y^2-z^3).$$ Necessary optimality conditions in the field have the form: $$\begin{cases} f'_y = (1-y^2-z^3)^2-4y^2(1-y^2-z^3)+2yz-2yz^2 = 0\\ f'_z = -6yz^2(1-y^2-z^3)+y^2+2z(1-y^2-z^3)-3z^4 = 0, \end{cases}$$ or $$\begin{cases} 5y^4+y^2(6z^3-6)+y(2z-2z^2)+(z^3-1)^2 = 0\\ 6y^3z^2+y^2(1-2z)+6y(z^3-1)z^2-5z^4+2z = 0. \end{cases}$$

If to consider the coefficient of the highest power of $$y$$ as denominator in equation, and the remaining coefficients - numerators, we get the above equations. Then we can subtract the second equation factor $$y$$ from the first and repeat subtraction with factor $$1$$, obtaining the system

$$\begin{cases} C_{2,2}(z)y^2 + C_{2,1}(z)y + C_{2,0}(z) = 0\\ 6y^3z^2+y^2(1-2z)+6y(z^3-1)z^2-5z^4+2z = 0, \end{cases}$$ where $$C_{2,2}(z) = 36z^7-36z^4+20z^2-20z+5,$$ $$C_{2,1}(z) = 18z^6+102z^5-30z^2,$$ $$C_{2,0}(z) = 36z^{10}-72z^7+50z^5+11z^4-20z^2+10z.$$

Thus, the order of the first equation in $$y$$ reduced from fourth to second. Likewise, lowering the order of the second equation by a first equation, we obtain

$$\begin{cases} C_{2,2}(z)y^2 + C_{2,1}(z)y + C_{2,0}(z) = 0\\ D_{2,2}(z)y^2 + D_{2,1}(z)y + D_{2,0}(z) = 0,\\ \end{cases}\qquad(1)$$ where $$D_{2,2}(z) = 180z^8+576z^7-72z^5-144z^4+40z^3-60z^2+30z-5,$$ $$D_{2,1}(z) = 180z^7-30z^6-30z^5-60z^3+30z^2,$$ $$D_{2,0}(z) = 180z^{11}-252z^8+100z^6-28z^5+25z^4-40z^3+40z^2-10z.$$

The system $$(1)$$ is a linear in the unknowns $$y^2$$ and $$y$$, so $$y^2=\dfrac{\Delta_2(z)}{\Delta_0(z)},\quad y=\dfrac{\Delta_1(z)}{\Delta_0(z)},\qquad(2)$$ where $$\Delta_0(z) = C_{2,2}(z)D_{2,1}(z) - C_{2,1}(z)D_{2,2}(z),$$ $$\Delta_2(z) = -C_{2,0}(z)D_{2,1}(z) + C_{2,1}(z)D_{2,0}(z),$$ $$\Delta_1(z) = -C_{2,2}(z)D_{2,0}(z) + C_{2,0}(z)D_{2,2}(z),$$ or $$\Delta_0(z) = 90z^{10}-828z^9-1662z^8-144z^7+396z^6+1028z^5-200z^4+180z^3-220z^2+10z,$$ $$\Delta_2(z) = -90z^{13}+540z^{12}+30z^{11}+234z^{10}-864z^9-290z^8+256z^7+74z^6+220z^5-160z^4+100z^3-50z^2,$$ $$\Delta_1(z) = 576z^{13}-1296z^{10}+90z^9+815z^8+444z^7-5z^6-580z^5+126z^4+10z^3+80z^2-30z-5.$$

From $$(2)$$ for $$\Delta_0(z)\not=0$$ should be $$\Delta_1^2(z) - \Delta_2(z)\Delta_0(z) = 0,$$ or $$331776z^{26}-1484892z^{23}-19440z^{22}+1233720z^{21}+3079404z^{20}+195732z^{19}-3189924z^{18}-3109428z^{17}+368233z^{16}+2734116z^{15}+1243978z^{14}-741000z^{13}-805907z^{12}+75696z^{11}+164040z^{10}+82560z^9-172194z^8+53440z^7-10290z^6+32440z^5-7460z^4-4400z^3+100z^2+300z+25 = 0.$$

The coefficients are particially calculated using the Mathcad package, and $$\mathcal{polyroots}()$$ function is also used, which calculates all the roots of the polynomial by the "accompanying matrix" method.

Calculating values $$y$$ with $$(2)$$ and $$x,f$$ by the formula $$x = 1-y^2-z^3,\quad f=xy^2+yz^2+zx^2$$ and checking them by substituting in the original system, we obtain the following stationary points with the positive coordinates: $$(x,y,z,f)\in\left[\genfrac{}{}{0pt}{0} {(0.5581125,\ 0.4984120,\ 0.5783713,\ 0.4856221)} {(0.1636702,\ 0.7619816,\ 0.6347238,\ 0.4548812)} \right.$$

• Less time consuming way to set out as a separate answer. – Yuri Negometyanov May 27 '16 at 12:43
• The problem with this is that the equations were solved using some solver (approximations) and so cant represent exact solution – ibnAbu May 27 '16 at 18:49
• Worst there is no proof that the approximate solutions give global maximum. – ibnAbu May 27 '16 at 18:50
• @stalker2133 Approximation seems a single way to get exact maximum, because we've got a hard task. "Classic" (clear) proof is shown there in another my answer, it was in my link. But never note it.))) – Yuri Negometyanov May 27 '16 at 19:44
• I checked the link but I wonder why we are using determinant to tell about solutions of non- linear equations. – ibnAbu May 27 '16 at 21:16

My new proof (2020/12/06)

Since $$y^2 + \frac{1}{4} \ge y$$, it suffices to prove that $$x^2(y^2 + \tfrac{1}{4}) + y^2 z + z^2 x < \frac{1}{2}.$$ Since $$y^2 = 1-x-z^3$$, it suffices to prove that $$4x^3+(4z^3-5)x^2+(-4z^2+4z)x+4z^4-4z+2 > 0. \tag{1}$$

If $$x = 0$$, (1) is true since $$4z^4-4z+2=(2z^2-1)^2+(2z-1)^2$$. (due to @Yuri Negometyanov)

If $$x > 0$$, by noting that \begin{align} &4x^3+(4z^3-5)x^2+(-4z^2+4z)x+4z^4-4z+2\\ =\ & 4x\left(x + \frac{1}{2}z^3 - \frac{11}{16}\right)^2 + \frac{1}{2}\left(x - 2\sqrt{2z^4 - 2z + 1}\right)^2\\ &\quad + x\left[2\sqrt{2z^4 - 2z + 1} - \left(z^6 - \frac{11}{4}z^3 + 4z^2 - 4z + \frac{121}{64}\right)\right], \tag{2} \end{align} it suffices to prove that, for all $$z \in [0, 1]$$, $$2\sqrt{2z^4 - 2z + 1} - \left(z^6 - \frac{11}{4}z^3 + 4z^2 - 4z + \frac{121}{64}\right) > 0. \tag{3}$$ Since $$2z^4 - 2z + 1 - (1 - \frac{3}{4}\sqrt[3]{2}) = \frac{1}{8}(3\sqrt[3]{4} + 4\sqrt[3]{2} z + 4z^2)(2z - \sqrt[3]{2})^2$$, we have $$\sqrt{2z^4 - 2z + 1} \ge \sqrt{1 - \frac{3}{4}\sqrt[3]{2}} > \frac{19}{81}$$. We have $$\sqrt{2z^4 - 2z + 1} = 1 - \frac{2z - 2z^4}{1 + \sqrt{2z^4 - 2z + 1}} \ge 1 - \frac{2z - 2z^4}{1 + \frac{19}{81}}.$$ It suffices to prove that, for all $$z \in [0, 1]$$, $$2 - \frac{4z - 4z^4}{1 + \frac{19}{81}} - \left(z^6 - \frac{11}{4}z^3 + 4z^2 - 4z + \frac{121}{64}\right) > 0$$ or $$-1600z^6 + 5184z^4 + 4400z^3 - 6400z^2 + 1216z + 175 > 0.$$ We have \begin{align} &-1600z^6 + 5184z^4 + 4400z^3 - 6400z^2 + 1216z + 175\\ \ge\ & -1600z^6 + 5184z^4 + 4400z^3 - 6400z^2 + 1216z + 175 - 1600(1 - z^2)(z^2-\tfrac{1}{5})^2\\ =\ & 2944z^4 + 4400z^3 - 5696z^2 + 1216z + 111\\ >\ & 2\sqrt{2944z^4 \cdot 111} + 2\sqrt{4400z^3\cdot 1216z} - 5696z^2\\ =\ & (2\sqrt{2944\cdot 111} + 2\sqrt{4400\cdot 1216} - 5696)z^2\\ \ge\ & 0. \end{align} We are done.

Remark: Actually, (1) is true for all $$x\ge 0, z\ge 0$$.

• $4z^4-4z+2=(2z^2-1)^2+(2z-1)^2$ – Yuri Negometyanov Dec 6 '20 at 0:09
• @YuriNegometyanov Very nice! Thanks. – River Li Dec 6 '20 at 0:21
• And try $18\times(1) - (6x+2z^3-5/2)\times(1)',$ this is correct in the stationary point. – Yuri Negometyanov Dec 6 '20 at 0:47
• @YuriNegometyanov Thanks. Which one is $(1)'$? – River Li Dec 6 '20 at 1:17
• Partial derivative of LHS(1) by $x.$ – Yuri Negometyanov Dec 6 '20 at 1:23

Let $A=x+y^2$, $B=y^2+z^3$ and $C=z^3+x$.

Claim: $x^2y \leq \dfrac{A}{2}x^{3/2}$, $y^2z \leq \dfrac{B^{2/3}}{4^{2/3}}y^{4/3}$, $z^2x \leq \dfrac{C^{4/3}}{4^{2/3}}x^{1/3}$.

Assuming the claim, we note that $A+B+C=2$ and we need to prove that the given sum is less than $\dfrac{A+B+C}{4}$.

• Isn't a+b+c=2, and that would change the answer – avz2611 May 24 '16 at 15:18
• Thanks, I'll modify the proof. – Aravind May 24 '16 at 18:00
• so assuming those three additional inequalities, can you at least prove the requested inequality? If not what progress does this represent? – hkBst May 27 '16 at 10:28

The method to be employed here is exactly the same as in a previous answer:

Let $u = x$ , $v = y^2$ , $w = z^3$ , then $u,v,w \ge 0$ , $u+v+w=1$ and the inequality to be established: $$x^2y+y^2z+z^2x < \frac12 \quad \Longrightarrow \quad f(u,v,w) = u^2v^{1/2}+v\,w^{1/3}+w^{2/3}u < \frac12$$ The maximum of this function inside the equilateral triangle must shown to be less than $1/2$.
There is no symmetry argument, because the latter is effectively destroyed by the "weird" condition $x+y^2+z^3=1$ , as it is called. Another proof without words is attempted by plotting a contour map of the function, as depicted. Levels (nivo) of these isolines are defined (in Delphi Pascal) as:

nivo := min + sqrt(g/grens)*(max-min); { sqrt = square root ; grens = 25 ; g = 0..grens }

The darkness of the isolines is proportional to the (positive) function values; they are almost black near the maximum and almost white near the minimum values. Maximum and minimum values of the function are observed to be:

 4.58251457205350E-0003 < f < 4.85621276951755E-0001 < 1/2

The little $\color{blue}{\mbox{blue}}$ spot is where $\left|f(u,v,w) - \mbox{max}\right|< 0.0002$ . This maximum is close to values found by other people here, but not quite. Perhaps it's interesting to know the position of the maximum as well:

(x,y,z) = ( 5.58304528246164E-0001 , 4.97693736095187E-0001 , 5.78892473099889E-0001 )

I have no idea how to convert these numerical values into something more analytical.

EDIT. What you see is what you get :-) Without doubt. Some decent error analysis reveals that the value of the maximum to be trusted in this answer is : $\;0.48562 \pm 0.00003\;$ , quite in agreement with values given elsewhere (e.g. in the comment by Steve Kass).

I don't think the best bound can be solved for analytically.

using Lagrange multipliers , I tried to maximize $x^2y+y^2z+z^2x$ $\quad$ subject to the constraint $x+y^2+z^3=1$

maximize $g=x^2y+y^2z+z^2x+\lambda(x+y^2+z^3-1)$

$\frac {\partial g}{\partial x}=0:2xy+z^2+\lambda=0$

$\frac {\partial g}{\partial y}=0:x^2+2yz+2\lambda y=0$

$\frac {\partial g}{\partial z}=0:y^2+2zx+3\lambda z^2=0$

$\frac {\partial g}{\partial \lambda}=0:x+y^2+z^3-1=0$

$x\frac {\partial g}{\partial x}=0:2x^2y+z^2x+\lambda x=0$

$y\frac {\partial g}{\partial y}=0:x^2y+2y^2z+2\lambda y^2=0$

$z\frac {\partial g}{\partial z}=0:y^2z+2z^2x+3\lambda z^3=0$

$3(x^2y+y^2z+z^2x)=-\lambda (x+2y^2+3z^3)=(2xy+z^2)(x+2y^2+3z^3)$

If the global maximum occur on $x, y, z \ge 0$ then we can solve $(2xy+z^2)(x+2y^2+3z^3) \ge 0$ for $x, y, z \ge 0$

If no local maximum occur at $x, y,z \ge 0$ then we have to search for farthest points from the point of global minimum.

• It looks like all we have to do is search for the farthest point away from the local minimum that also satisfies the constraint. – ibnAbu May 24 '16 at 7:58

Let $x=0$.

Then $y^2+z^3=1 \Rightarrow z^3=1-y^2$

Seek to maximise $p=y^2z=y^2(1-y^2)^{\frac 13}$

WLOG let $u=y^2$ so that $p=u(1-u)^{\frac 13}$

$\frac {dp}{du}=(1-u)^{\frac 13}-u{\frac 13}(1-u)^{-\frac 23}$

$\frac {dp}{du}=\frac 13(3-4u)(1-u)^{-\frac 23}$

Maximum is at $u=\frac 34$

Thus maximum is $p=\frac 34(\frac 14)^{\frac 13}$

$p \le \frac 34(\frac 14)^{\frac 13}$

$p^3 \le \frac {27}{64}\frac 14$

$p^3 \le \frac {27}{256} < \frac {32}{256}$

$p^3 < \frac {1}{8}$

$p < \frac {1}{2}$

We then have to demonstrate that any increase in $x$ creates corresponding decreases in $y^2$ and $z^2$ such that the other two terms in the expression $x^2y+y^2z+z^2x$ increase more slowly than the decrease in $p$.

• You find a maximum at $p = \frac{3}{4}\left(\frac{1}{4}\right)^{\frac{1}{3}} \approx 0.47247$. However, in the comments to the question, Ewan Delanoy found a point which has value $0.484818$. So I don't think your last paragraph will work? – sTertooy May 24 '16 at 12:07
• I was hoping someone else would address that... – tomi May 24 '16 at 12:12
• It also seems odd that the maximum I found is at $\frac {\sqrt 3} 2$. Is there some trig transformation that would make this easier? – tomi May 24 '16 at 12:14
• @stalker2133 look there: math.stackexchange.com/questions/1775498/… – Yuri Negometyanov May 27 '16 at 18:26

$$\color{green}{\textbf{Computer free version of 17.12.20.}}$$

$$\color{brown}{\textbf{Preliminary notes.}}$$

We should prove the inequality$$\;f(x,y,z)= x^2y+y^2z+z^2x \le \frac12\;$$ under the conditions $$x+y^2+z^3=1,\;$$ $$x,y,z\in [0,1].$$

In accordance with the @RiverLi answer,

• $$y\le y^2+\frac14,\tag1$$
• $$2-4(x^2y+y^2z+z^2x)\ge 2-x^2-(4x^2+4z)(1-x-z^3)-4z^2x,\tag2$$

where $$(1)$$ follows from the evident $$\;\left(y-\frac12\right)^2 \ge0\;$$.

At the same time, for $$\;t\in[0,1]\;$$ $$t^4-t+\frac3{4\sqrt[\large3]4} = \left(t^2-\sqrt[\large3]2t-\frac1{2\sqrt[\large3]2}\right)\left(t^2 +\sqrt[\large3]2 t +\frac3{2\sqrt[\large3]2}\right)$$ $$= \left(t-\frac1{\sqrt[\large3]4}\right)^2\left(t^2+\sqrt[\large3]2 t +\frac3{2\sqrt[\large3]2}\right) \ge 0,$$

$$t\le t^4+H,\quad \text{where}\quad H=\frac3{4\sqrt[\large3]4} = \dfrac12\sqrt[\large3]{\dfrac{27}{32}}<\dfrac12,\quad H\approx0.47247.\tag3$$

$$\color{brown}{\textbf{The edges of area.}}$$

Taking in account $$(1)$$ and $$(3),$$ one can get \begin{align} &f(x,y,0) = x^2y \le (1-y^2)^2y\,\bigg|_{\large y=\frac1{\sqrt5}} = \dfrac{16}{25\sqrt5}<0.28622, \\[4pt] &f(x,0,z) = z^2x = (1-z^3)z^2 \le (1-z^3)z \le H, \\[4pt] &f(0,y,z) = y^2z = (1-z^3)z \le H. \end{align}

Thus, the given inequality is correct on the edges of the area.

$$\color{brown}{\textbf{The inner stationary points.}}$$

Taking in account $$(2)$$ and $$(1)$$ in the form of $$\;4z\le 4z^2+1,\;$$ it suffices to prove the inequality $$\;g(x,z)\ge0,\;$$ where $$g(x,z) = 2-x^2-(4x^2+1+4z^2)(1-x-z^3)-4z^2x,$$ $$g(x,z) = 1+4x^2z^3 + (4z^5+z^3-4z^2) + (4x^3-5x^2+x),\tag4$$ under the conditions $$\;\;x,z\in(0,1).$$

Stationary points should satisfy the system $$\;g'_x = g'_z =0,\;$$ or $$\begin{cases} g'_x = 8z^3x + 12 x^2 - 10x + 1 = 0\\[4pt] g'_z = 12x^2z^2+20 z^4+3z^2-8z = 0,\tag5 \end{cases}$$

and then $$\begin{cases} 12x^2z = 8-3z-20z^3\hspace{163mu} \left[z^{-1}\times(5.2)\right]\\ 8z^4x +(8-3z-20z^3) - 10xz + z = 0,\qquad \left[z\times(5.1)\right]\\ \end{cases}$$ $$x = h_1(z) = \dfrac{4-z-10z^3}{5z-4z^4},\quad x^2 = h_2(z) = \dfrac{8-3z-20z^3}{12z}\tag6.$$

At the same time, from $$(4),(5)$$ should \begin{align} &5g(x,z) = 5g - xg'_x - z g'_z\\[4pt] & = 5+20x^2z^3 + (20z^5+5z^3-20z^2) + (20x^3-25x^2+5x)\\[4pt] & - x(8z^3x + 12 x^2 - 10x + 1) - z(12x^2z^2+20 z^4+3z^2-8z)\\[4pt] & = 5 + 4 x + 8x^3 -15x^2 - 12 z^2 + 2 z^3, \end{align} $$5g(x,z) = 5 - 15h_2(z) + 4 h_1(z) (1+2h_2(z)) - 12 z^2 + 2 z^3.\tag7$$ Substitution of $$(6)$$ to $$(7)$$ represents the inequality $$\;g(x,z) \ge 0\;$$ in the form of $$256 - 568 z + 501 z^2 - 1280 z^3 + 1180 z^4 - 300 z^5 + 1600 z^6 - 624 z^7 - 96 z^8\ge 0.\tag8$$

Since $$h(z)-1=\dfrac{4z^4-10z^3-6z+4}{5z-4z^4}=\dfrac{2(1-2z)(2+z+2z^2-z^3)}{5z-4z^4},$$ then $$\;h(z)>1,\;$$ if $$\;z\in\left(0,\frac12\right).\;$$

Besides, $$\;h(z)\;$$ has a positive denominator and decreasing numerator in the interval $$\;z\in(0,1],\;$$ wherein $$\;h\left(\frac7{10}\right) = -\frac{325}{6349}<0.\;$$ Therefore, $$\;h(z)< 0,\;$$ if $$\;z\in\left[\frac7{10},1\right].\;$$

Thus, all inner stationary points belong to the interval $$\;z\in\left[\dfrac12,\dfrac7{10}\right).\;$$

And now, substitution $$\;z=\dfrac{6+t}{10},\;t\in[-1,1),\;$$ transforms $$(8)$$ to the evident inequality $$-3 t^8 - 339 t^7 - 6214 t^6 - 13083 t^5 + 1041140 t^4\\ + 12923432 t^3 + 64494426 t^2 - 898904 t + 13278632 \ge 0.$$

Done!

• 1) Should it be $f(x,0,z) = z^2x = (1-z^3)z^2 \le (1-z^3)z \le H, f(0,y,z) = y^2z = (1-z^3)z \le H$? – River Li Dec 18 '20 at 3:32
• 2) How do you get the 3rd equation in (5)? How do you get the equation between (5) and (6)? More details should be given. – River Li Dec 18 '20 at 3:32
• 3) Also, the 1st equation in (4) is different from RHS of (2)? – River Li Dec 18 '20 at 3:41
• 4) Check (6) $x = \cdots$ – River Li Dec 18 '20 at 12:31
• The sentence around "has a vertical asymptote at", maybe the details should be given how to get $h(z) > 1$? Also, the next paragraph... – River Li Dec 18 '20 at 15:35

More idea than answer, but it might be of help.

If we replace $x$ by $x^2$ and $z$ by $z^{2/3}$, the constraint becomes $x^2+y^2+z^2=1$ and the inequality becomes

$$x^4y+y^2z^{2/3}+z^{4/3}x^2\lt{1\over2}$$

This looks worse, but the advantage of constraining $(x,y,z)$ to the unit sphere is that we can switch over to spherical coordinates, writing $x=\sin\theta\cos\phi$, $y=\sin\theta\sin\phi$, and $z=\cos\theta$, and make use of some trig identities. Ultimately, with a final change of variable $u=(\cos\theta)^{2/3}$ and $v=-\cos2\phi$, the inequality to prove (if I've done the algebra correctly) is

$$(1-u^2)^{5/2}\left({(1-v)(1+v)^4\over8}\right)^{1/2}+(u-u^4)(1+v)+(u^2-u^5)(1-v)\lt1\quad\text{for }0\le u,v\le1$$

This "reduces" the problem to the unpleasant chore of finding (or bounding) a function of two variables over the unit square. I'll end by noting that if we focus on the easy portion of the function, $f(u,v)=(u-u^4)(1+v)+(u^2-u^5)(1-v)$, we find that $f$ is maximized when $v=1$ and $u=1/\sqrt[3]4$, and

$$f\left({1\over\sqrt[3]4},1\right)={3\over2\sqrt[3]4}\approx0.94494$$

so the true minimum of the full function is not much less than $1$. (Just a note: I multiplied both sides of the original inequality by $2$ to clear out some denominators. Dividing by $2$ changes $0.94494$ to $0.47247$, which is, as it should be, less than the numerical result $0.48562209920309984177$ reported by Steve Kass in comments below the OP.)