This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$

let $0<a\le b\le c\le d$, and such $abcd=1$,show that $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$

it seems harder than This inequality $a+b^2+c^3\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}$

Solution 1:

To take account of the restriction $0 \le a \le b \le c \le d$ we define three variables $u,v,w$ which will each be restricted to $[1,\infty)$ and put $b=au^4,\ c=bv^4,\ d=cw^4.$ The reason for the fourth powers is to have simple representations for $a,b,c,d$ which make them also satisfy the restriction $abcd=1.$ This is an iff type condition which then leads to unique expressions for each of $a,b,c,d$ namely $$ a=\frac{1}{u^3v^2w} \\ b=\frac{u}{v^2w} \\ c=\frac{uv^2}{w} \\ d=uvw^2.$$

Now let $f(a,b,c,d)=a-1/a+b^2-1/b^2+c^3-1/c^3+d^4-1/d^4.$ the expression we wish to show is nonnegative. We substitute in $f$ the expressions above for $a,b,c,d$ in terms of $u,v,w$ and get an equivalent expression $g(u,v,w)$ in the new variables. The advantage so far is that there are now no restrictions on $u,v,w$ other than that each is in $[1,\infty)$

Using a CAS we obtain for $g(u,v,w)$ an expression having denominator $u^4v^8w^{12}$ and its numerator is $m(u,v,w)$ where $$m(u,v,w)=u^8v^{16}w^{24}+(v^4-w^4)[u^7v^{10}w^9+uv^2w^{11}] \\ +u^6v^4w^{10}-u^2v^{12}w^{14}-1.$$ If now it happens that $v \ge w$ the value of $m$ is easily seen to be nonnegative, since (recall $u,v,w \ge 1$) the first term dominates the variable subtracted one, and also clearly $u^6v^4w^{10}-1 \ge 0.$

The remaining case is that in which $v<w,$ and I could see no slick way to complete this case. However it is indeed provable by defining (after changing the variables for no good reason but my notes) the polynomial $h(x,y,t)=m(x,y,y+t).$ So here $t$ is measuring the extent by which the third of $u,v,w$ exceeds the second, to reflect we're looking at the remaining case $v<w.$ When this $h(x,y,t)$ is expanded into a polynomial in $t$ of degree 24, with coefficients each polynomials in $x$ and $y$, it is fairly easy to see these coefficients are each nonnegative, though admittedly tedious. For example the coefficient of $t^4$ is $$xy^{10}(10626x^7y^{26}-589x^6y^9+210x^5-1001xy^{12}-1035y^3).$$ Here the first term dominates the subtracted ones in terms of each of its exponents exceeding those of other terms, and also (rather clearly in this case) the coefficient contribution from the first term more than offsets that of the subtracted terms. It goes the same for all the other terms, I just don't know if anyone wants me to type it all in (I'd rather not).

I'd like to see a slicker proof at least of the case $v<w.$

[I deleted a proposed simplification since to show a polynomial has nonnegative derivatives involves showing all of its (the derivative's) coefficients nonnegative, which is basically done as above with no easy short-cut.]

Solution 2:

it is trivial $d \ge 1$

there are two cases: $c\ge 1$ , or $c\le 1$

$c\ge 1$

let $x=\dfrac{1}{a},y=\dfrac{1}{b} \implies x\ge y\ge 1 ,xy=cd$

let $f(x)=x-\dfrac{1}{x}$, it is trivial $f(x)$ is mono increasing

function

now we need to prove :

$f(c^3)+f(d^4) \ge f(x)+f(y^2)$

1)if $ c^3< x \cap c^3<y^2 \implies yc^3 <xy=cd \iff d> c^2y>y \cap x^2c^3 <x^2y^2=c^2d^2 \iff d^2>cx^2 \iff d>x $

$f(d^4)=f(d^2)(d^2+\dfrac{1}{d^2}) \ge 2 f(d^2) > f(x^2)+f(y^2)>f(x) +f(y^2)$

$f(c^3) \ge 0$ and this part is true.

2)if $x>y^2 \cap y^2 <c^3<x \implies c^3d^3=x^3y^3 <xd^3 \iff d^3>y^3x^2 \iff d^4>x^2 $

$f(c^3)+f(d^4) >f(y^2)+f(x^2)>f(x)+f(y^2)$

3)if $x<y^2 \cap x<c^3<y^2 \implies x^3y^3=c^3d^3 <y^2d^3\iff d^3>x^2y \ge y^3 \iff d>y$

$f(c^3)>f(x), f(d^4)>f(y^4)>f(y^2)$ it is true also.

4)it is trivial true when $c^3>x,c^3>y^2$

when $c \le 1 ,z=\dfrac{1}{c} \implies x \ge y \ge z \ge 1, d=xyz \ge 1$

we need to prove : $ f(d^4) \ge f(x)+f(y^2)+f(z^3)$

$f(z^4) \ge f(z^3),\iff f((xyz)^4) \ge f(x)+f(y^2)+f(z^4) \\ \iff (xy)^4((xy)^4-1)v^2+x^2y^2(x^2+xy^2-x^3y^2-x^2y^4)v+(xy)^4-1 \ge 0 ,v=z^4$

$(xy)^4((xy)^4-1)>0,$ if $\Delta \le 0 $ then it is true.

$\Delta \le 0 \\ \iff (x^2+xy^2-x^3y^2-x^2y^4)^2 \le 4((xy)^4-1)^2 \\ \iff x^2y^4+x^3y^2-x^2-xy^2 \le 2((xy)^4-1) \\ \iff (2x^4-x^2)y^4-(x^3-x)y^2+x^2-2 \ge 0 $

$g(t=y^2)=(2x^4-x^2)t^2-(x^3-x)t+x^2-2$

note $\dfrac{(x^3-x)}{2(2x^4-x^2)}<1 \iff 4x^3 -x^2 -2x+1 \ge 0 \iff 2x^2(2x-1)+(x-1)^2 \ge 0$ is true,

$g_{min}=g(1)=2x^4-x^3+x-2=(x^2-1)(2x^2-x+2) \ge 0$