Generalized Cauchy difference functional equations. (English) Zbl 1079.39017

The article concerns primarily the study of functional equations of the form \[ f(x)+ f(y)- f(x+ y)= g(H(x,y)), \] where \(H\) is a given of two (real or complex) variables and where \(f\) and \(g\) are unknown functions. Frist, the author studies the special case \[ f(x)+ f(y)- f(x+ y)= h(\phi(x)+ \phi(y)- \phi(x+y)), \] \(\phi\) analytic, whose solution is of the form \[ h(s)= A_1(1)+ c,\;s\in \{\phi(x)+ \phi(y)- \phi(x+ y)\mid (x,y)\in I^2\}, \] for \(f(x)= A_1\circ \phi(x)+ A_2(x)+c\) where \(A_1\) and \(A_2\) are arbitrary additive maps and \(c\) is a constant, \(I\) being a nonempty open connected subset of the field \(\mathbb{R}\) or \(\mathbb{C}\).
Further, the author studies the general case when \[ H(x,y)= \psi(\phi(x)+ \phi(y)- \phi(x+ y)). \] Under suitable conditions the solution is the following \[ g\circ\psi(u)= A_1(u)+ c,\;u\in \{\phi(x)+ \phi(y)- \phi(x+ y)\mid x,y\in I\}, \] for \(f(x)= A_1\circ\phi(x)+ A_2(x)+ c\), \(x\in I\). Many consequences and examples are given. The proofs are exhaustive.


39B22 Functional equations for real functions
26A51 Convexity of real functions in one variable, generalizations
39B32 Functional equations for complex functions
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Full Text: DOI