The standard Young's inequality states, for $a,b>0$ and $\nu\in[0,1]$, that
$$ \nu a + (1-\nu)b\geq a^{\nu}b^{1-\nu}. $$
This has been refined in various ways, for example see the refined Young's inequality, which states that
$$ \nu a + (1-\nu)b\geq a^\nu b^{1-\nu}K(b/a,2)^{\min(\nu, 1-\nu)} $$ where
$$K(t, 2) = \frac{(t+1)^2}{4t}$$
satisfies $K(t,2) = K(t^{-1},2)$, so $K(b/a,2) = K(a/b,2)$, and $K(t, 2)\geq 1$ for all $t>0$, so this inequality is sharper.
These type of arguments can lead to refinements of Holders inequality $\lVert f,g\rVert \leq \lVert f\rVert_p\lVert g\rVert_q$ for $p^{-1}+q^{-1} = 1$ and both $p,q$ finite. I am curious if anything is known about the case one of $p,q = \infty$.
Essentially, I need an upper bound on $\langle \vec x, \vec y\rangle$ in terms of the $\ell_1$ norm of $\vec y$. I can clearly do this with $\langle \vec x, \vec y\rangle \leq \lVert \vec x\rVert_\infty \lVert \vec y\rVert_1$, but I just learned about the refined Young's inequality (and it just helped me move from a trivial bound to a non-trivial bound elsewhere), so I am curious if there are refined bounds in my setting.