Convexity of the Petrović-Type Functionals

and Applied Analysis 3 Remark 1.5. If f x /x is increasing on I and 1.6 holds, then the Petrović-type functional P1 is nonnegative, that is, inequality 1.4 is valid. Conversely, if f x /x is increasing on I and conditions as in 1.7 are fulfilled, then relation 1.5 holds. In order to define another Petrović-type functional, we cite the following Petrović-type inequality involving a convex function. Theorem 1.6. Let I 0, a ⊆ R , x1, . . . , xn ∈ I and let p1, . . . , pn ∈ R fulfill conditions as in 1.1 . If f : I → R is a convex function, then f ( n ∑ i 1 pixi ) ≥ n ∑ i 1 pif xi ( 1 − n ∑ i 1 pi ) f 0 . 1.8 Remark 1.7. If f is a concave function then −f is convex, hence replacing f by −f in Theorem 1.6, we obtain inequality