On the order of convergence of some sequences of Riemann’s sums of functions with convex derivative

Let / c.R be an open interval. f:1-4•R a differentiable function with f : 1 R convex,. and a, b c/, a . h. hi this note, by using a Taylor-type result tegurdmg left-hand derivative. we prove the following formula , 1 limn II a + (2i I) • f (.0fix — —   ■ n “”. 2n 24 which is wellArtown for an other class of functions, see (IL Part IL Chap. 1, 02. namely for f twice differentiable with f integrahle on [alit.