ABSTRACT
Q-class Function Inequalities (a) If f is of class Q on [a, b] then for all x, y ∈ [a, b] and λ, 0 < λ < 1,
0 ≤ f((1− λ)x + λy) ≤ f(x) 1− λ +
f(y)
λ . (1)
(b) If f is of class Q on [a, b] and if a ≤ xi ≤ b, 1 ≤ i ≤ 3 then f(x1)(x1−x2)(x1−x3)+f(x2)(x2−x3)(x2−x1)+f(x3)(x3−x1)(x3−x2) ≥ 0. (c) If f is of class Q on [a, b] and if a ≤ x1 < x2 < x3 ≤ b then
f(x2) ≤ x3 − x1 x3 − x2 f(x1) +
x3 − x1 x2 − x1 f(x3);
0 ≤ f(x1) x3 − x2 +
f(x2)
x1 − x3 + f(x3)
x2 − x1 .
