Jensen's type inequalities for the finite Hilbert transform of convex functions

In this paper we obtain some new inequalities for the finite Hilbert transform of convex functions by the use of Jensen’s integral inequality. Applications for exponential function are provided as well.

Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization of $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

Inequalities for the finite Hilbert transform of functions with bounded divided differences

In this paper we establish some inequalities for the finite Hilbert transform of complex valued functions for which the divided differences in any two points of the interval are bounded. Applications for some particular functions of interest are provided as well.