  This paper deals with the initial-boundary value problem of the biharmonic
cubic nonlinear Schr\"odinger equation in a quarter plane with inhomogeneous
Dirichlet-Neumann boundary data. We prove local well-posedness in the low
regularity Sobolev spaces introducing Duhamel boundary forcing operator
associated to the linear equation to construct solutions on the whole line.
With this in hands, the energy and nonlinear estimates allow us to apply
Fourier restriction method, introduced by J. Bourgain, to get the main result
of the article. Additionally, adaptations of this approach for the biharmonic
cubic nonlinear Schr\"odinger equation on star graphs are also discussed.
