1
$\begingroup$

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

$\endgroup$
3
$\begingroup$

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!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.