How to prove that $x^{1-x}+(1-x)^{x}\leq x^{1/2}+(1-x)^{1/2}$?
I use https://eudml.org/doc/223938 , pdf-file page $135$, section $7$ with $\enspace a:=1-x\enspace $ and $\enspace b:=x$ .
Because of the symmetry of the inequality $\enspace x^{1-x}+(1-x)^x\leq \sqrt{x}+\sqrt{1-x}\enspace $ at $\enspace x=0.5$ ,
see $\enspace x\to 1-x$, it’s enough to check the mentioned inequality for $\enspace 0\leq x\leq 0.5$
instead of $\enspace 0\leq x\leq 1$ .
We have to proof for $\enspace 0\leq x\leq 0.5$ :
$(1.1) \hspace{1cm}\displaystyle x^{1-x}\leq\sqrt{4x^2(1-x)+2x(1-x)(1-2x)\ln(1-x)}$
$(1.2) \hspace{1cm}\displaystyle \sqrt{4x^2(1-x)+2x(1-x)(1-2x)\ln(1-x)}\leq\sqrt{x}(1-(1-x)(1-2x)\ln 2)$
$(1.3) \hspace{1cm}\displaystyle -4+8x^2+(8x-6)(1-x)(1-2x)\ln 2<4x-2+(8x-6)\ln(2(1-x))$
$(1.4) \hspace{1cm}\displaystyle x(3-2x)(3-4x)\ln 2<1$
$(2.1) \hspace{1cm}\displaystyle (1-x)^x\leq\sqrt{1-4x^2(1-x)-2x(1-x)(1-2x)\ln(1-x)}$
$(2.2) \hspace{1cm}\displaystyle \sqrt{1-4x^2(1-x)-2x(1-x)(1-2x)\ln(1-x)}\leq\sqrt{1-x}(1+x(1-2x)\ln 2)$
$(2.3) \hspace{1cm}\displaystyle (1-4x)(\ln 2)^2-\frac{2}{1-x}<-1\leq\frac{3-4x}{(1-x)^2(1-2x)^2}-\frac{4}{1-2x}$
$(3) \hspace{1.3cm}\displaystyle \sqrt{x}(1-(1-x)(1-2x)\ln 2)+\sqrt{1-x}(1+x(1-2x)\ln 2)\leq \sqrt{x}+\sqrt{1-x}$
The proofs are carried out from the bottom upwards.
To $(3)$ :
$0\leq x\leq 0.5\enspace $ => $\enspace 0\leq x\leq 1-x\enspace $ => $\enspace \sqrt{x}\leq\sqrt{1-x}\enspace $ => $\enspace x\sqrt{1-x}\leq (1-x)\sqrt{x}$
=> $\enspace x\sqrt{1-x}(1-2x)\ln 2\leq (1-x)\sqrt{x}(1-2x)\ln 2$
=> $\enspace \sqrt{x}(-(1-x)(1-2x)\ln 2)+\sqrt{1-x}(x(1-2x)\ln 2)\leq 0$
=> $\enspace \sqrt{x}(1-(1-x)(1-2x)\ln 2)+\sqrt{1-x}(1+x(1-2x)\ln 2)\leq \sqrt{x}+\sqrt{1-x}$
To $(2.3)$ :
On the one hand it’s $\enspace (1-x)(1-4x)(\ln 2)^2\leq (\ln 2)^2<1\leq 1+x\enspace $ and therefore
$\displaystyle (1-4x)(\ln 2)^2- \frac{2}{1-x}<-1\enspace $ and on the other hand because of $\enspace x\leq 0.5\enspace $ we get
$(a)\hspace{1cm}x^2(-1+2x)\leq 0\enspace$ => $\enspace (1-x)^2(3+2x)\leq 3-4x\enspace\enspace$ and
$(b)\hspace{1cm}(1-x)^2(1-2x)(3+2x)\leq (1-x)^2(3+2x)$ .
With $\enspace (a) \enspace $ and $\enspace (b) \enspace $ follows $\enspace (1-x)^2(1-2x)(3+2x)\leq 3-4x$
and therefore $\enspace\displaystyle -1\leq \frac{3-4x}{(1-x)^2(1-2x)^2}-\frac{4}{1-2x}$ .
To $(2.2)$ based on $(2.3)$ :
It’s $\enspace\displaystyle \frac{d}{dx}(-2+x(1-2x)(\ln 2)^2+2\ln(2(1-x)))=(1-4x)(\ln 2)^2-\frac{2}{1-x}\enspace$ and
$\displaystyle \frac{d}{dx}(\frac{1}{(1-x)(1-2x)}-\frac{2}{1-2x})=\frac{3-4x}{(1-x)^2(1-2x)^2}-\frac{4}{1-2x}\enspace$ and it follows
$\displaystyle \frac{d}{dx}(-2+x(1-2x)(\ln 2)^2+2\ln(2(1-x)))<\frac{d}{dx}(\frac{1}{(1-x)(1-2x)}-\frac{2}{1-2x})$ .
Together with the common point
$\displaystyle (-2+x(1-2x)(\ln 2)^2+2\ln(2(1-x)))|_{x=0.5}=-2=(\frac{1}{(1-x)(1-2x)}-\frac{2}{1-2x})|_{x=0.5}\enspace $ does it mean that
$-2+x(1-2x)(\ln 2)^2+2\ln(2(1-x)) \enspace $ and $\enspace\displaystyle \frac{1}{(1-x)(1-2x)}-\frac{2}{1-2x}\enspace $ don’t touch each other for $\enspace 0\leq x<0.5$ .
Taking a value of this value range, e.g. $x=0$, we get
$\displaystyle (\frac{1}{(1-x)(1-2x)}-\frac{2}{1-2x})|_{x=0}=-1<$
$\displaystyle <-2+2\ln 2=(-2+x(1-2x)(\ln 2)^2+2\ln(2(1-x)))|_{x=0}$
and therefore $\enspace\displaystyle \frac{1}{(1-x)(1-2x)}-\frac{2}{1-2x}\leq -2+x(1-2x)(\ln 2)^2+2\ln(2(1-x))$ .
Elementary transformations lead to
$1-4x^2(1-x)-2x(1-x)(1-2x)\ln(1-x)\leq (1-x)(1+x(1-2x)\ln 2)^2$ .
To $(2.1)$ : $\enspace$ That’s formula $(7.2)$, see the link above.
To $(1.4)$ :
With $\enspace\displaystyle \frac{d}{dx}(x(3-2x)(3-4x)\ln 2)=((x-\frac{3}{4})^2-\frac{3}{16})24\ln 2:=0\enspace $ follows
$\displaystyle x_{1,2}=\frac{3\pm\sqrt{3}}{4}\enspace $ and with $\enspace\displaystyle \frac{d^2}{dx^2}(x(3-2x)(3-4x)\ln 2)=(2x-\frac{3}{2})24\ln 2<0$
for $\enspace 0\leq x\leq 0.5\enspace $ follows concavity of $\enspace x(3-2x)(3-4x)\ln 2$ .
Therefore we get
$ x(3-2x)(3-4x)\ln 2\leq\max(x(3-2x)(3-4x)\ln 2)=$
$=(x(3-2x)(3-4x)\ln 2)|_{x=(3-\sqrt{3})/4}=0.75\sqrt{3}\ln 2<1$ .
To $(1.3)$ based on $(1.4)$ :
It’s $\enspace 2x<2x+1-x(3-2x)(3-4x)\ln 2\enspace $ and with
$4x^2\leq 2x\enspace $ we get $\enspace 4x^2<2x+1-x(3-2x)(3-4x)\ln 2$ .
Elementary transformations lead to
$4x^2+(4x-3)(1-x)(1-2x)\ln 2<2x+1+(4x-3)\ln 2$ .
Because of $\enspace 4x-3<0\enspace $ and $\enspace \ln(2(1-x))\leq \ln 2$
we have $(4x-3)\ln 2\leq (4x-3)\ln(2(1-x))\enspace$ and it follows
$4x^2+(4x-3)(1-x)(1-2x)\ln 2<2x+1+(4x-3)\ln 2\leq$
$\leq 2x+1+(4x-3)\ln(2(1-x))$
and with elementary transformations using the left and right side we get
$-4+8x^2+(8x-6)(1-x)(1-2x)\ln 2<4x-2+(8x-6)\ln(2(1-x))$ .
To $(1.2)$ based on $(1.3)$ :
It’s
$\displaystyle \frac{d}{dx}(1-4x(1-x)+((1-x)(1-2x)\ln 2)^2)=-4+8x^2+(8x-6)(1-x)(1-2x)\ln 2$
and
$\displaystyle \frac{d}{dx}(2(1-x)(1-2x)\ln(2(1-x)))=4x-2+(8x-6)\ln(2(1-x))$
so that we get
$\displaystyle \frac{d}{dx}(1-4x(1-x)+((1-x)(1-2x)\ln 2)^2)<\frac{d}{dx}(2(1-x)(1-2x)\ln(2(1-x)))$ .
Together with the common point
$(1-4x(1-x)+((1-x)(1-2x)\ln 2)^2)|_{x=0.5}=0=(2(1-x)(1-2x)\ln(2(1-x)))|_{x=0.5}\enspace $
does it mean that
$1-4x(1-x)+((1-x)(1-2x)\ln 2)^2\enspace $ and $\enspace 2(1-x)(1-2x)\ln(2(1-x))$
don’t touch each other for $\enspace 0\leq x<0.5$ .
Taking a value of this value range, e.g. $x=0$, we get
$(1-4x(1-x)+((1-x)(1-2x)\ln 2)^2)|_{x=0}=1+(\ln 2)^2>$
$>2\ln 2=(2(1-x)(1-2x)\ln(2(1-x)))|_{x=0}$
and therefore $\enspace 2(1-x)(1-2x)\ln(2(1-x))\leq 1-4x(1-x)+((1-x)(1-2x)\ln 2)^2$.
Elementary transformations lead to
$4x^2(1-x)+2x(1-x)(1-2x)\ln(1-x)\leq x(1-(1-x)(1-2x)\ln 2)^2$ .
To $(1.1)$: $\enspace$ That’s formula $(7.1)$, see the link above.
Now we have
$x^{1-x}+(1-x)^x$
$\leq\sqrt{4x^2(1-x)+2x(1-x)(1-2x)\ln(1-x)}$
$\hspace{0.5cm}+\sqrt{1-4x^2(1-x)-2x(1-x)(1-2x)\ln(1-x)}$
$\leq\sqrt{x}(1-(1-x)(1-2x)\ln 2)+\sqrt{1-x}(1+x(1-2x)\ln 2)$
$\leq\sqrt{x}+\sqrt{1-x}$ .