General and alien solutions of a functional equation and of a functional inequality

The purpose of the present paper is to solve (under some assumption on the domain) the equation $g(x+y)-g(x)-g(y)=xf(y)+yf(x).$ After determining the general solutions, we will investigate the so--called alien solutions. %More precisely, we will examine the cases when the above equation implies that %$g(x+y)=g(x)+g(y)$ and $xf(y)+yf(x)=0$. %Concerning this, necessary and sufficient conditions will be provided. Finally, we will discuss the real solutions of the following related functional inequality: $g(x+y)-g(x)-g(y)\geq xf(y)+yf(x).$

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