Norm of the Hilbert matrix on Bergman spaces

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It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space Ap into Ap if and only if 2<p<∞. In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space Ap is equal to πsin⁡2πp, when 4≤p<∞, and it was also conjectured that‖H‖Ap→Ap=πsin⁡2πp, when 2<p<4. In this paper we prove this conjecture.