On Some Trigonometric Functional Inequalities

We deal with d’Alembert’s and Wilson’s differences $$f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right)f\left( y \right)$$ and $$f\left( x \right)f\left( y \right) - f{\left( {\frac{{x + y}}{2}} \right)^2} + f{\left( {\frac{{x - y}}{2}} \right)^2}$$ respectively, assuming that their absolute values (or norms) are majorized by some function in a single variable. The superstability type results obtained are then used to characterize the functions $$x \mapsto \;\cos \alpha x\;{\text{and}}\;x \mapsto \;\sin \;\alpha x$$ of real variable x with a complex but not real coefficient α.