OpenStudy (anonymous):

Let $x _{i}>0$ for i=1,2,3,...,n. For each positive integer k, prove that: $(x _{i}^{k}+...+x _{n}^{k})/n≤(x _{i}^{k+1}+...x _{n}^{k+1})/(x_{1}+x_{2}+...+x_{n})$

6 years ago
OpenStudy (anonymous):

Rewrite the equation as $\frac{x_1+\cdots+x_n}{n} \le \frac{x_1^{k+1}+\cdots +x_n^{k+1}}{x_1^k +\cdots + x_n^k}$ Note that this is the same as $\frac{x_1^1+\cdots + x_n^1}{x_1^0+\cdots +x_n^0}\le \frac{x_1^{k+1}+\cdots +x_n^{k+1}}{x_1^k +\cdots + x_n^k}$ There is a type of mean known as the Lehmer mean, which is $L_a=\frac{x_1^a+\cdots +x^a_n}{x^{a-1}_1+\cdots x^{a-1}_n}$ Using a bit of calculus, namely taking the derivative with respect to a, you'll known that L_a is a monotone increasing function with respect to a. Because it is increasing, you have that $p \le q \implies L_p \le L_q$ Note that your inequality is the same as $L_1 \le L_k$ which (using Lehmer's inequality) is obviously true for all k greater than or equal to one, which encompasses all positive integers.

6 years ago
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