simple arithmetic inequality

Question

Im looking for some advice in this question. I know about the Hoelder Inequalities for $1\leq p \leq \infty$, but I did not figure out a way to apply this to $ \sum a_{k} \leq C\big(\sum a_{k}^{1/p}\big)^{p}$ for $a_k\ge 0, C > 0$. Any help is appreciated. Thanks

Answer 1

If $a_1=1$ and $a_2=...=a_n=0$ then we get $C\geq1$ and the inequality $$\left(\sum\limits_{k=1}^na_k^{\frac{1}{p}}\right)^p\geq\sum\limits_{k=1}^na_k$$ follows from Karamata.

Indeed, let $a_1\geq a_2\geq...\geq a_n$ and $a_k^{\frac{1}{p}}=x_k$.

Thus, we need to prove that $$(x_1+x_2+...+x_n)^p\geq x_1^p+x_2^p+...+x_n^p$$ and since $f(x)=x^p$ is a convex function and $$(x_1+x_2+...+x_n,0,...,0)\succ(x_1,x_2,...,x_n),$$ we are done!