I know that Holder's inequality already states, for non-negative sequences ${a(n)},{b(n)}$, that $$((∑a(n)b(n))/((∑a^{p}(n))^{1/p}(∑b^{q}(n))^{1/q}))≤1$$ where $p>1$ and $(1/p)+(1/q)=1$ and Reversed Holder states that $$((∑a(n)b(n))/((∑a^{1/p}(n))^{p}(∑b^{1/(1-p)}(n))^{1-p}))≥1$$ where $p>1$. My question is, Is there a non-trivial lower bound of $((∑a(n)b(n))/((∑a^{p}(n))^{1/p}(∑b^{q}(n))^{1/q}))$ and upper bound of $((∑a(n)b(n))/((∑a^{1/p}(n))^{p}(∑b^{1/(1-p)}(n))^{1-p}))$??
p.s. I think those bounds might be dependent on the power $p$.
p.s. If there is no such thing, Is there any helpful tips on how to prove and obtain one?