July 31, 2010

(Van Khea):Problem 026

Filed under: Uncategorized — KKKVVV @ 9:08 am

Let a, b, c, k>0; a\leq b\leq c. Prove that:

\displaystyle \frac{a^k}{b}+\frac{b^k}{c}+\frac{c^k}{a}\geq \frac{b^k}{a}+\frac{c^k}{b}+\frac{a^k}{c}


Let \displaystyle f(x)=\frac{1}{x^{\frac{1}{k}}} ; k>0

\displaystyle f''(x)=\frac{1}{k}(\frac{1}{k}+1)x^{-\frac{1}{k}-2}>0

From the conditional a\leq b\leq c ; k>0\Longrightarrow a^k\leq b^k\leq c^k

From the inequality by Van Khea for m=n=p\neq 0a^k\leq b^k\leq c^k and f''(x)>0 we get:

(c^k-b^k)f(a^k)-(c^k-a^k)f(b^k)+(b^k-a^k)f(c^k)\geq 0

\displaystyle \Longrightarrow \frac{c^k-b^k}{a}-\frac{c^k-a^k}{b}+\frac{b^k-a^k}{c}\geq 0

\Longrightarrow\displaystyle \frac{a^k}{b}+\frac{b^k}{c}+\frac{c^k}{a}\geq \frac{b^k}{a}+\frac{c^k}{b}+\frac{a^k}{c}

Therefor the proof is completed.

See also the inequality by VanKhea


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