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

TL;DR

This paper characterizes general and alien solutions to a specific functional equation and inequality, providing necessary and sufficient conditions for particular cases and exploring the structure of solutions under certain assumptions.

Contribution

It offers a comprehensive analysis of solutions to a functional equation and inequality, including the identification of alien solutions and conditions for their existence.

Findings

Derived the general solutions of the functional equation.

Identified conditions under which solutions are alien solutions.

Analyzed solutions of the related functional inequality.

Abstract

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: $$ g(x+y)-g(x)-g(y)\geq xf(y)+yf(x). $$