A NOTE ON FUNCTIONS WITH THE CONCEPT OF CONVEXITY

The tgs−convex functions are studied in many papers. A non-negative function f : I ⊆ R → R is called a tgs−convex function if the inequality f(tx + (1 − t)y) ≤ t(1 − t)(f(x) + f(y)) holds for all x, y ∈ I and t ∈ (0, 1). In this note we prove that every tgs−convex function is non-positive. So the only non-negative tgs−convex function is the zero function. So all the results about the non-negative tgs−convex functions are trivial.