Recently, I started to teach myself calculus by reading Apostol's Calculus Volum 1. I almost finished the introductory chapter. However, I felt the condition is imperfect for exercise 17 from section 4.10 of the introductory chapter.
Inequalities relating different types of averages. Let $x_1,x_2,\ldots,x_n$ be $n$ positive real numbers. If $p$ is a nonzero integer, the $p$th-power mean $M_p$ of the $n$ numbers is defined as follows: $$M_p=\left(\frac{x_1^p+\cdots+x_n^p}{n}\right)^{1/p}.$$ The number $M_1$ is also called the arithmetic mean, $M_2$ the root mean square, and $M_{-1}$ the harmonic mean.
17. If $p > 0$, prove that $M_p<M_{2p}$ when $x_1,x_2,\ldots,x_n$ are not all equal.
[Hint: Apply the Cauchy-Schwarz inequality with $a_k=x_k^p$ and $b_k=1$.]
I can construct a counterexample as follows. Let $x_1 = -2$, $x_2 = 2$, and $p = 2$, then $$ M_2 = \left(\frac{(-2)^2 + 2^2}{2}\right)^{1/2} = \left(\frac{4+4}{2}\right)^{1/2} = 4^{1/2} = 2 = 16^{1/4} = \left(\frac{16+16}{2}\right)^{1/2} = \left(\frac{(-2)^4 + 2^4}{2}\right)^{1/2} = M_4. $$ I believe that the condition should be “when $|x_1|, |x_2|, \dots, |x_n|$ are not all equal.” Is there anyone who can reassure me about my thinking?