MHB Can inequality be proven with positive real numbers and fractions?

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Inequality
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
For the positive real numbers $x,\,y$ and $z$ that satisfy $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=3$, prove that

$\dfrac{1}{\sqrt{x^3+1}}+\dfrac{1}{\sqrt{y^3+1}}+\dfrac{1}{\sqrt{z^3+1}}\le \dfrac{3}{\sqrt{2}}$.
 
Mathematics news on Phys.org
My solution:

Using cyclic symmetry, we find the extremum must occur for:

$$(x,y,z)=(1,1,1)$$

If we define:

$$f(x,y,z)=\frac{1}{\sqrt{x^3+1}}+\frac{1}{\sqrt{y^3+1}}+\frac{1}{\sqrt{z^3+1}}$$

then we obtain:

$$f(1,1,1)=\frac{3}{\sqrt{2}}$$

To determine the nature of the extremum, we pick another point on the constraint:

$$(x,y,z)=\left(2,\frac{4}{5},\frac{4}{5}\right)$$

And we find:

$$f\left(2,\frac{4}{5},\frac{4}{5}\right)=\frac{21+10\sqrt{105}}{63}<\frac{3}{\sqrt{2}}$$

Thus, the extremum is a maximum, and we may state:

$$f(x,y,z)\le\frac{3}{\sqrt{2}}$$
 
MarkFL said:
My solution:

Using cyclic symmetry, we find the extremum must occur for:

$$(x,y,z)=(1,1,1)$$

If we define:

$$f(x,y,z)=\frac{1}{\sqrt{x^3+1}}+\frac{1}{\sqrt{y^3+1}}+\frac{1}{\sqrt{z^3+1}}$$

then we obtain:

$$f(1,1,1)=\frac{3}{\sqrt{2}}$$

To determine the nature of the extremum, we pick another point on the constraint:

$$(x,y,z)=\left(2,\frac{4}{5},\frac{4}{5}\right)$$

And we find:

$$f\left(2,\frac{4}{5},\frac{4}{5}\right)=\frac{21+10\sqrt{105}}{63}<\frac{3}{\sqrt{2}}$$

Thus, the extremum is a maximum, and we may state:

$$f(x,y,z)\le\frac{3}{\sqrt{2}}$$

Good job, MarkFL! And thanks for participating!

My solution:

By AM-GM, we have $x^3+1\ge 2x\sqrt{x}$ so $\dfrac{1}{\sqrt{x^3+1}}\le\dfrac{1}{\sqrt{2}\sqrt{x}\sqrt[4]{x}}$. By the same token we also have $\dfrac{1}{\sqrt{y^3+1}}\le\dfrac{1}{\sqrt{2}\sqrt{y}\sqrt[4]{y}}$ and $\dfrac{1}{\sqrt{z^3+1}}\le\dfrac{1}{\sqrt{2}\sqrt{z}\sqrt[4]{z}}$.

Adding the three inequalities we get:

$\dfrac{1}{\sqrt{x^3+1}}+\dfrac{1}{\sqrt{y^3+1}}+\dfrac{1}{\sqrt{z^3+1}}\le\dfrac{1}{\sqrt{2}}\left(\dfrac{1}{\sqrt{x}\sqrt[4]{x}}+\dfrac{1}{\sqrt{y}\sqrt[4]{y}}+\dfrac{1}{\sqrt{z}\sqrt[4]{z}}\right)$

Note that the following can be obtained by Cauchy–Schwarz inequality:

$\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}}\le\sqrt{1+1+1}\sqrt{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}=\sqrt{3}\sqrt{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}$

$\dfrac{1}{\sqrt{x}\sqrt[4]{x}}+\dfrac{1}{\sqrt{y}\sqrt[4]{y}}+\dfrac{1}{\sqrt{z}\sqrt[4]{z}}\le\sqrt{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}\sqrt{\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}}}=\sqrt{3}\sqrt{\left(\sqrt{3}\sqrt{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}\right)}$

Putting these pieces together, and since we're told that $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=3$ we see that we have proved:

$\begin{align*}\dfrac{1}{\sqrt{x^3+1}}+\dfrac{1}{\sqrt{y^3+1}}+\dfrac{1}{\sqrt{z^3+1}}&\le\dfrac{1}{\sqrt{2}}\left(\dfrac{1}{\sqrt{x}\sqrt[4]{x}}+\dfrac{1}{\sqrt{y}\sqrt[4]{y}}+\dfrac{1}{\sqrt{z}\sqrt[4]{z}}\right)\\&\le \dfrac{1}{\sqrt{2}} \sqrt{3}\sqrt{\left(\sqrt{3}\sqrt{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}\right)}\\&\le \dfrac{3}{\sqrt{2}}\end{align*}$
 
anemone said:
Good job, MarkFL! And thanks for participating!...

You likely knew I would use cyclic symmetry just as surely as I knew you would use AM-GM. (Smirk)
 
MarkFL said:
You likely knew I would use cyclic symmetry just as surely as I knew you would use AM-GM. (Smirk)

LOL! That is very true!
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Back
Top