On continuous solutions of a generalized Gołab–Schinzel functional inequality

In the paper we improve some results from [9] concerning the Goła̧b–Schinzel inequality $$f(x+f(x)y)\le f(x)f(y)$$ for a continuous unknown function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ . We also show some properties of solutions of the generalized inequality $$f(x+g(x)y)\le f(x)f(y)$$ for continuous unknown functions $$f,g:\mathbb {R}\rightarrow \mathbb {R}$$ provided $$g\ge f$$ .