An Inequality with Just Two Variable And an Integer

Solution

From means the inequality

(1)

$\displaystyle\frac{a}{b^n}+\frac{b}{a^n}\geq 2\sqrt{\frac{ab}{a^nb^n}}.$

(2)

$\displaystyle\frac{a^n}{b^n}+\frac{b}{a}\geq 2\sqrt{\frac{a^nb}{b^na}}.$

(3)

$\displaystyle\frac{b^n}{a^n}+\frac{a}{b}\geq 2\sqrt{\frac{b^na}{a^nb}}.$

By multiplying the relationship (1),(2),(3):

(4)

$\displaystyle\Bigr(\frac{a}{b^n}+\frac{b}{a^n}\Bigr)\Bigr(\frac{a^n}{b^n}+\frac{b}{a}\Bigr)\Bigr(\frac{b^n}{a^n}+\frac{a}{b}\Bigr)\geq \frac{8}{\sqrt{a^{n-1}\cdot b^{n-1}}}.$

We prove that

(5)

$\displaystyle\frac{a^n}{b}+\frac{b^n}{a}\geq a^{n-1}+b^{n-1}.$

\begin{align} &a^{n+1}+b^{n+1}\geq a^nb+ab^n\\ &a^n(a-b)-b^n(a-b)\geq 0\\ &(a-b)(a^n-b^n)\geq 0\\ &(a-b)^2(a^{n-1}+a^{n-2}b+\ldots+b^{n-1})\geq 0, \end{align}

which is true. Now multiply the relationships (4),(5)

\displaystyle\begin{align} &\Bigr(\frac{a^n}{b}+\frac{b^n}{a}\Bigr)\Bigr(\frac{a}{b^n}+\frac{b}{a^n}\Bigr)\Bigr(\frac{a^n}{b^n}+\frac{b}{a}\Bigr)\Bigr(\frac{b^n}{a^n}+\frac{a}{b}\Bigr)\geq \frac{8(a^{n-1}+b^{n-1})}{\sqrt{a^{n-1}\cdot b^{n-1}}}\\ &\qquad\qquad=8\Biggl(\sqrt{(\frac{a}{b})^{n-1}}+\sqrt{(\frac{b}{a})^{n-1}}\Biggl) \end{align}

Acknowledgment

This problem, along with a solution, was kindly communicated to me by Dan Sitaru. Dan has eatlier published the problem at the Romanian Mathematical Magazine.

Inequalities in Two Variables