So here is a question from the book 'Problems in Mathematical Analysis by W.J. Kaczor and M.T. Nowak'. I tried to solve this problem and I'll attach my solution in this question. Please let me know whether my solution is correct or not. And most likely its not correct then I would like to request the reader to give a pure solution.
Question: Let ${a_1}$,${a_2}$,…,${a_p}$ be fixed positive numbers. Consider the sequence
$${s_n}=\frac{{a_1}^n+{a_2}^n+…+{a_p}^n}{p}$$ and ${x_n}={s_n}^\frac{1}{n}, n \in \Bbb N$.
Show that the sequence $\{x_n \}$ is monotonically increasing.
Best Answer
the function $f(x) = x^{(n + 1)/n}$ is convex then : $$\dfrac{f(a_1^n) + \cdots + f(a_p^n)}{p} \geq f \left(\dfrac{a_1^n + \cdots + a_p^n}{p}\right)$$ so : $$\dfrac{a_1^{n + 1} + \cdots + a_p^{n + 1}}{p} \geq \left(\dfrac{a_1^n + \cdots + a_p^n}{p}\right)^{(n + 1)/n}$$ We deduce that : $$s_{n + 1} \geq s_n^{(n + 1)/n}$$ and finally : $$x_{n + 1} = \left(s_{n + 1}\right)^{1/(n + 1)} \geq \left(s_n\right)^{1/n)} = x_n$$ the sequence $(x_n)$ is then increasing