Let f be continuous function in open interval I and a∈I.
We define
L_a (t)≔(f(t)+f(a))/2 (t-a)-∫_a^t〖f(s)ds,a≤t〗 。
P_a (t)≔∫_a^tf(s)ds-(t-a)f((t+a)/2),a≤t。
We study the properties of L_a and P_a.
[5] S. S. DRAGOMIR, Some integral inequalities for differentiable convex functions, Contributions Macedonian Acad. of Sci. and Arts (Macedonia) 13 (1), 13-17 (1992).