6

Let $ a, b> 0 $ and $\frac{1}{a}+\frac{1}{b}=1.$ Prove that$$\sqrt{a^2+4}+\sqrt{b^2+4}\leq\frac{\sqrt{2}}{4}(a+b)^2.$$

Obviously $a+b=ab\ge4$. Other than that, I do not know what trick to use to deal with the constraint. The Lagrange multiplier method has not gotten me further either.

Hans
  • 10,484

5 Answers5

4

Let $x=\frac{1}{a}$ and $y=\frac{1}{b}$. We want to show that $x+y=1$ and $x,y\geq 0$ imply

$$ y^2x\sqrt{4x^2+1}+x^2y\sqrt{4y^2+1}\leq \frac{\sqrt{2}}{4}. $$

Letting $x=\frac{1+t}{2},y=\frac{1-t}{2}$, this is equivalent to finding the maximum of $$ f(t) =(1-t^2)\left[ (1-t)\sqrt{1+(1+t)^2}+(1+t)\sqrt{1+(1-t)^2}\right]$$ (which is an even function) over $[-1,1]$. We have $$ f(t) = (1-t^2)\sqrt{4-2t^2+2t^4+2(1-t^2)\sqrt{1+t^4}} $$ $$ f(\sqrt{t}) = \sqrt{2}(1-t)\sqrt{2-t+t^2+(1-t)\sqrt{1+t^2}}$$ and both $1-t$ and $2-t+t^2+(1-t)\sqrt{1+t^2}$ are positive and decreasing functions over $[0,1]$, so the maximum of $f(t)$ is attained at the origin. Indeed $$2-\tan\theta+\tan^2\theta+(1-\tan\theta)\sqrt{1+\tan^2\theta} = \frac{2+\sqrt{2}\sin\left(\tfrac{\pi}{4}+\theta\right)}{2\sin^2\left(\frac{\pi}{4}+\frac{\theta}{2}\right)}$$ and the derivative of the RHS is $$ -\frac{1}{\sin^2\left(\frac{\pi}{4}+\frac{\theta}{2}\right)\left(1+\tan\frac{\theta}{2}\right)}<0.$$

Jack D'Aurizio
  • 361,689
3

By AM-GM and C-S we obtain: $$\frac{\sqrt2(a+b)^2}{4}=\frac{\sqrt{2(a+b)}\sqrt{a+b}(a+b)^2}{4ab}\geq(\sqrt{a}+\sqrt{b})\sqrt{a+b}=$$ $$=\sqrt{a^2+ab}+\sqrt{b^2+ab}\geq\sqrt{a^2+\frac{4a^2b^2}{(a+b)^2}}+\sqrt{b^2+\frac{4a^2b^2}{(a+b)^2}}=$$ $$=\sqrt{a^2+4}+\sqrt{b^2+4}.$$

Hans
  • 10,484
Michael Rozenberg
  • 203,855
2

To get rid of some square roots and simplify, substitute $(a,b)\to (2x, 2y)$. The inequality can be written as:

Let $x,y >0$ and $\frac{1}{x}+\frac{1}{y} = 2$. Prove that:

$$\sqrt{2(x^2+1)}+\sqrt{2(y^2+1)} \leq (x+y)^2$$

In this case, from AM-GM we can see that:

$$\sqrt{2(x^2+1)}+\sqrt{2(y^2+1)} \leq \frac{x^2+1+2}{2}+\frac{y^2+1+2}{2} = \frac{x^2+y^2}{2}+3$$

It will be enough to prove:

$$\frac{x^2+y^2}{2}+3 \leq (x+y)^2$$

or

$$x^2+y^2+4xy \geq 6$$

From AM-GM on the initial condition, we can see that $xy \geq 1$. Therefore

$$x^2+y^2+4xy \geq 6xy \geq 6$$

Equality occurs when $(x,y) = (1,1)$, so when $(a,b)=(2,2)$.

LHF
  • 1
2

You can show it just by squaring, using GM-AM and applying the two facts

  • $(1)$: $a+b = ab$ and
  • $(2)$: $ab \geq 4$

LHS: $$\left(\sqrt{a^2+4}+\sqrt{a^2+4}\right)^2 =a^2+b^2+8 + 2\sqrt{(a^2+4)(b^2+4)}\stackrel{GM-AM}{\leq}2(a^2+b^2+8)$$

RHS: $$\left(\frac{\sqrt 2}{4}(a+b)^2\right)^2 \stackrel{(1)}{=} \frac 18 (a+b)^2(ab)^2 \stackrel{(2)}{\geq}2(a^2+b^2+2ab)$$ $$ \stackrel{(2)}{\geq}2(a^2+b^2+8)$$

trancelocation
  • 33,349
  • 1
  • 21
  • 46
1

Hint: You can use the constraint $\frac{1}{a}+\frac{1}{b}=1$ to

  1. solve $a$ in terms of $b$ and substitute
  2. define a function in one variable $ f(b) \geq 0$ $$\cases{f(b) = \frac{\sqrt 2}{4}(a(b)+b)^2-\sqrt{a(b)^2+4}-\sqrt{b^2+4}\\\text{ where}\\ a(b) = \left(1-\frac{1}{b}\right)^{-1} = \frac{b}{b-1}}$$
  3. do one variable calculus on $f$ w.r.t. the remaining variable $b$.

If we do this, we can verify $f(2)=f'(2)=0$. The reason for why to calculate at 2 is that we can realize that $2$ is special just with symmetry, that $a=b=2$ should be special, since both constraint and inequality are symmetric w.r.t. a,b. $f(b) = f(\frac{b}{b-1})$.

What remains is to prove sub-interval $]1,2]$ (or $[2,\infty[$) is monotonic.

Here we should have done Jack's substitutions as it makes the calculations easier.

mathreadler
  • 26,534
  • 1
    Well, have you tried solving the resulting high order polynomial equation from setting the first derivative to zero in this brute force way? – Hans Jan 24 '20 at 17:34
  • The main point was to prove that the symmetric point $a=b=2$ was local minimum. Then we can with Jack's substitution prove that either side will be monotonic. – mathreadler Jan 24 '20 at 19:32
  • What is $f$? Why does $f$ minimize at a single symmetric point $(2,2)$ rather than two two points symmetric to each other $(a, b)$ and $(b,a)$? – Hans Jan 24 '20 at 19:36
  • $b\to f(b)$ is $RHS-LHS$ with the substitution described. The symmetry in the original expression, you should see that it is equal if you switch places of all the $a$s and $b$s. So the point of symmetry on the curve is where $b=\frac{b}{b-1}$, which happens when $b=2$. – mathreadler Jan 24 '20 at 19:41
  • We understand what the symmetry point you are referring to. My point is that you have neither defined what $f$ is nor proved why the symmetric point $(2,2)$ minimizes $f$. – Hans Jan 24 '20 at 19:58