A sharp Blaschke–Santaló inequality for α-concave functions

We define a new transform on α-concave functions, which we call the ♯-transform. Using this new transform, we prove a sharp Blaschke–Santaló inequality for α-concave functions, and characterize the equality case. This extends the known functional Blaschke–Santaló inequality of Artstein-Avidan, Klartag and Milman, and strengthens a result of Bobkov. Finally, we prove that the ♯-transform is a duality transform when restricted to its image. However, this transform is neither surjective nor injective on the entire class of α-concave functions.