On differentiable and monogenic functions in a harmonic algebra

Abstract

For locally bounded and differentiable in the sense of G\^ateaux functions $\Phi$ given in a three-dimensional commutativeharmonic algebra with two-dimensional radical, we prove the following statement: if the function $\Phi$ domain is convex "in the radical direction" and the difference $\zeta_1-\zeta_2$ belongs to the radical, the difference $\Phi(\zeta_1)-\Phi(\zeta_2)$ belongs also to the radical. As a result, we prove that locally bounded and differentiable in the sense of G\^ateaux functions are also differentiable in the sense of Lorch.