Document Type : Research Articles

Author

10.22080/cjms.2021.20097.1559

Abstract

Let f be a convex function on I and a, b∈I with a
0 ≤(1/2)∫_{a}^{b}|t-((a+b)/2)|p(t)dt[f₊′(((a+b)/2))-f₋′(((a+b)/2))]
≤∫_{a}^{b}p(t)f(t)dt-(∫_{a}^{b}p(t)dt)f(((a+b)/2))
≤(1/2)∫_{a}^{b}|t-((a+b)/2)|p(t)dt[f₋′(b)-f₊′(a)]

and

0 ≤(1/2)∫_{a}^{b}[(1/2)(b-a)-|t-((a+b)/2)|]p(t)dt[f₊′(((a+b)/2))-f₋′(((a+b)/2))]
≤(∫_{a}^{b}p(t)dt)((f(a)+f(b))/2)-∫_{a}^{b}p(t)f(t)dt
≤(1/2)∫_{a}^{b}[(1/2)(b-a)-|t-((a+b)/2)|]p(t)dt[f₋′(b)-f₊′(a)].

Keywords