On a Cubic Equation and a Jensen-Quadratic Equation

Abstract

We obtain the general solutions of the cubic functional equation $\text{3}\left[g\left(x+y\right)+g\left(x-y\right)+6g\left(x\right)\right]=2g\left(2x+y\right)+2g\left(2x-y\right)+g\left(-x-y\right)+g\left(-x+y\right)+6g\left(-x\right)$ and the Jensen-quadratic functional equation $f\left(\left(x+y\right)/2,z+w\right)+f(\left(x+y\right)/2,z-w)=f\left(x,z\right)+f\left(x,w\right)+f\left(y,z\right)+f\left(y,w\right)$.