On algebraic approximations of certain algebraic numbers

Using hypergeometric functions and the Thue–Siegel method we give an effective improvement of Liouvilleʹs approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalitiesBX4−AX3Y−6BX2Y2+AXY3+BY4 Nfor A 58B3 or A 308B4.