  We explicitly solve a variational problem related to upper bounds on the
optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT)
inequalities, which has recently been derived in [Invent. Math. 231 (2023),
no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math.
Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ]. We
achieve this through a variational characterization of the $L^1$ norm of the
Fourier transform of a function and duality, from which we obtain a
reformulation in terms of a variant of the Hadamard three lines lemma. By
studying Hardy-like spaces of holomorphic functions in a strip in the complex
plane, we are able to provide an analytic formula for the minimizers, and use
it to get the best possible upper bounds for the optimal constants in the CLR
and LT inequalities achievable by the method of [Invent. Math. 231 (2023),
no.1, 111-167. https://doi.org/10.1007/s00222-022-01144-7 ] and [J. Eur. Math.
Soc. (JEMS) 23 (2021), no.8, 2583-2600. https://doi.org/10.4171/jems/1062 ].
