Books on functional equations
You can find recent results of functional equations in the following references.
Marek Kuczma, An Introduction to the Theory of Functional Equations and Inequalities (Cauchy’s Equation and Jensen’s Inequality) Second Edition, ISBN 978-3-7643-8748-8 Birkhäuser Verlag
P. Kannappan, Functional equations and inequalities with applications, ISBN 978-0-387-89491-1 Springer
Stefan Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, ISBN 981-02-4837-7