We consider the optimal recovery of the βth degree of the Laplacian value on a function from the information on its κ-plane transform, measured with an error. We present the error of the optimal recovery and the set of optimal methods on classes with the bounded ath degree of the Laplacian, where 0 ≤ β < α. As a consequence, we give one inequality for the norms of the degree of the Laplace operator and the κ-plane transform. Particular cases include new inversion methods and inequalities for the classical Radon and x-ray transforms. © 2016 IOP Publishing Ltd Printed in the UK.