Hölder's inequality for finite sums is given by $$\sum_{k=0}^n|a_kb_k|\leq\left(\sum_{k=0}^n|a_k|^p\right)^{1/p}\left(\sum_{k=0}^n|b_k|^q\right)^{1/q},$$ where $1/p+1/q=1$, $p,q\in(1,\infty)$.
Is there a "similar" inequality which gives a lower bound for the left hand sum? I have searched, but found nothing so far.
On
Just consider Cauchy-Schwarz. We see that \begin{align} |a_1b_1 + a_2b_2| \leq \sqrt{a_1^2+a_2^2}\sqrt{b_1^2+b_2^2}. \end{align} If you are hoping for estimates of the nature \begin{align} f\left(\sqrt{a_1^2+a_2^2}, \sqrt{b_1^2+b_2^2}\right) \leq |a_1b_1+a_2b_2| \end{align} for some function $f(x, y)$ which is nonnegative on $\{x \geq 0, y\geq 0\}$, then you better hope if $(a_1, a_2)\perp (b_1, b_2)$ you have \begin{align} f\left(\sqrt{a_1^2+a_2^2}, \sqrt{b_1^2+b_2^2}\right) \leq 0. \end{align} Well. Guess $f$ has to be trivial.
the corresponding lower bound is $$ \max(|a_k b_k|) $$ which is not very interesting because it squelches $p$ and $q$, but it saturates so we can't produce a more interesting lower bound.