  The nonlinear Hausdorff-Young inequality follows from the work of Christ and
Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen
independently of the exponent. We show that the nonlinear Hausdorff-Young
quotient admits an even better upper bound than the linear one, provided that
the function is sufficiently small in the $L^1$ norm. The proof combines
perturbative techniques with the sharpened version of the linear
Hausdorff-Young inequality due to Christ.
