Proving a complicated looking inequality in a simple way

Question

This is again a search for alternative proofs:

Let $0 <s \le 1$, and suppose that $0 <a,b $ satisfy $$ ab=s,a+b=1+\sqrt{s}. \tag{1}$$

I have a proof for the assertion

$$ 2(1-\sqrt s)^3 \le |a-1|^3+|b-1|^3, \, \, \, \text{for every } \, s \ge \frac{1}{9}$$

but it is rather involved.

Actually, I am sure that the lower bound of $ \frac{1}{9}$, is not tight; the inequality holds for some $s > s^*$ where $s^* < \frac{1}{9}$.

Define $F(x,y):=|x-1|^3+|y-1|^3$. My proof is based on finding the global minimum $ \min_{xy=s} F(x,y)$.

However, here we need to show "only that" $F(\sqrt s, \sqrt s) \le F(a,b)$ for the specific $a,b$ described above in $(1)$.

Is there a way to prove this inequality "directly", without solving the harder global optimization problem?

Bonus: Is there a natural way to find the exact threshold $s^*$?

Edit:

There are now some very nice answers. I still wonder whether one can prove this without solving explicitly the quadratic described implicitly in $(1)$.

Here is an elementary proof for when $s \ge \frac{4}{9}$:

Suppose that $a \ge b$. The conditions on $a,b$ easily imply that $a \ge 1$, hence $s=ab \ge b$. Thus, we have $$ b \le s \le \sqrt s \le 1 \le a.$$

So, replacing $b$ with $\sqrt s$ clearly lowers the value of $F$, since we get closer to $1$. Now it is beneficial to replace $a$ by $\sqrt s$ when

$$|\sqrt s -1|=1-\sqrt s \le a-1 \iff 2-\sqrt s \le a \iff 4-2\sqrt s \le 2a. \tag{2}$$

Solving explicitly the quadratic $ a^2-(1+\sqrt s)a+s=0$, we get (assuming $a \ge b$) that

$$ a=\frac{1}{2}(1+\sqrt s+\sqrt{1+2\sqrt s-3s}).$$ Thus, inequality $(2)$ beceoms

$$ 4-2\sqrt s \le 1+\sqrt s+\sqrt{1+2\sqrt s-3s}, $$

or $3-3\sqrt s \le \sqrt{1+2\sqrt s-3s}$. Squaring this and simplifying gives

$$ 3s-5\sqrt s +2 \le 0, $$ which holds exactly for $\frac{4}{9} \le s \le 1$.

Answer 1

Let $\sqrt{s}=t$.

Thus, $\frac{1}{3}\leq t\leq 1.$

Since $a$ and $b$ are roots of the equation $$x^2-(1+t)x+t^2=0,$$ we need to prove that $$\left|\frac{1+t+\sqrt{(1+t)^2-4t^2}}{2}-1\right|^3+\left|\frac{1+t-\sqrt{(1+t)^2-4t^2}}{2}-1\right|^3\ge2(1-t)^3$$ or $$\left|\frac{\sqrt{(1-t)(1+3t)}-(1-t)}{2}\right|^3+\left|\frac{\sqrt{(1-t)(1+3t)}+(1-t)}{2}\right|^3\ge2(1-t)^3$$ or $$\left(\sqrt{1+3t}-\sqrt{1-t}\right)^3+\left(\sqrt{1+3t}+\sqrt{1-t}\right)^3\ge16\sqrt{(1-t)^3}$$ or $$\sqrt{(1+3t)^3}+3(1-t)\sqrt{1+3t}\ge8\sqrt{(1-t)^3}$$ or $$(1+3t)^3+6(1-t)(1+3t)^2+9(1-t)^2(1+3t)\ge64(1-t)^3$$ or $$4t^3-12t^2+15t-3\ge0,$$ which is true even for $t\ge\frac{1}{4}:$ $$4t^3-12t^2+15t-3=4t^3-t^2-11t^2+\frac{11}{4}t+\frac{49}{4}t-\frac{49}{16}+\frac{1}{16}=$$ $$=(4t-1)\left(t^2-\frac{11}{4}t+\frac{49}{16}\right)+\frac{1}{16}>0.$$

Answer 2

By symmetry, we may take $a\leqslant b$ without loss of generality. Since the upper bound on $s$ ensures that $a$ and $b$ cannot both exceed $1$, we have $a\leqslant1$. It is convenient to transform the variables as follows: $$t:=1-\surd s,\qquad u:=1-a,\qquad v:=b-1.$$ Then the relationships between $a$, $b$, and $s$ become $$u-v=t,\qquad uv=t-t^2.$$Clearly $u\geqslant v$ and $u\geqslant0$. Also $v\geqslant0$ since $uv=t-t^2\geqslant0$ for $0\leqslant t<1$. Hence $u$ and $-v$ are the roots in $x$ of the quadratic equation $$x^2-tx+t^2-t=0,$$ where $$u=\tfrac12\surd(4t-3t^2)+\tfrac12t,\qquad v=\tfrac12\surd(4t-3t^2)-\tfrac12t.$$ Let $$f(t):=u^3+v^3-2t^3.$$ Our task is to find the range of $t$ for which $f(t)\geqslant0$. Since $u^3+v^3=(u+v)[(u+v)^2-3uv]$, we have $$f(t)=t\surd(4t-3t^2)-2t^3.$$ In the given range $0\leqslant t<1$, this function initially increases from zero, attains its maximum $\frac14(\surd5-1)$ at $t=\frac12$, and then decreases to zero at $t=\alpha$, where $\alpha$ is the real root of $$t^3=1-\tfrac34t.$$ The corresponding value of $s$ is $(1-\alpha)^2,$ or approximately $0\!\cdot\!059354279$ according to my calculator.

Answer 3

The primary objective of this text is to give the following insight into this interesting issue : it boils down to the fact (see figure below) that a certain curve is above a certain straight line, boundary of the domain defined by homogenous inequation (4).

I will make the following change of variables similar to what @John Bentin has done : $$x:=a-1, \ \ \ y:=b-1, \ \ \ t:=\sqrt{s} \ \text{with} \ 0<t<1, \ \tag{1}$$

transforming the initial constraints into

$$\begin{cases}x+y&=&t-1\\ \ \ \ xy&=&t(t-1)\end{cases}\tag{2}$$

As $x$ and $y$ play a symmetrical rôle, we may assume $x<y$.

Using (2), a rapid calculation shows that (solving quadratic equation $X^2-(t-1)X+t(t-1)=0$) :

$$\begin{cases}x&=&\frac12(t-1-\sqrt{\Delta})\\y&=&\frac12(t-1+\sqrt{\Delta})\end{cases} \ \ \ \text{with} \ \Delta=(3t+1)(1-t)>0\tag{3}$$

It is immediate to see that $x<0$ whereas $y>0$ due to the second relationship in (2).

Therefore, the inequality we have to establish

$$2(1-t)^3 \le |x|^3+|y|^3$$

can be written

$$-2(x+y)^3 \leq -x^3+y^3\tag{4}$$

Consider now the plane with coordinates $(x,y)$. Let us plot in it,

• the curve $(C_1)$ (in red) with parametric equations (3).

• the (frontier) curve $(C_2)$ (in blue) with implicit equation (4) in which the $\leq$ sign has been replaced by the $=$ sign :

$$-2(x+y)^3 = -x^3+y^3\tag{5}$$

Fig. 1 : Representation of curves defined by (3) and (5). Please note that only the left hand side plane $x<0$ makes sense here.

Curve $(C_2)$ is a line. Not so surprizing in fact (see remark 1 below). Indeed, plugging $y=ux$ into (4) gives the following constraint on $u$ :

$$-2((1+u)x)^3 = (u^3-1)x^3 \ \ \iff \ \ -2(1+u)^3=u^3-1,\tag{6}$$

a third degree equation whose unique real root is $u_0 \approx -0.20406$, meaning that the equation of the line is approximately $y=-0.2x$.

Now, that we have well understood the nature of the frontier, we can infer that the region defined by inequation (4) is the half-plane situated above the straight line we have found (one reason among others : point $(x,y)=(0,1)$ belongs to this region).

It remains to prove, as suggested by the figure, that the red curve is entirely situated into this favorable region.

Remarks:

1) The fact that the curve associated with (5) is a straight line can be explained differently by considering that it's homogeneous (if $(x,y)$ is on the curve, $(\lambda x, \lambda y)$ belongs as well to the curve) ; technically speaking, we could have as well divide its LHS and RHS by $x^3$, generating a 3rd degree equation with variable $u:=\tfrac{y}{x}$.

2) (on an experimental basis) one can take $t \ge 0.2436...$ instead of $t \ge 1/3$.