ON CONNECTIONS BETWEEN APPROXIMATE 2ND-ORDER DIRECTIONAL DERIVATIVE AND 2ND-ORDER DINI DERIVATIVE FOR CONVEX-FUNCTIONS

For a real-valued convex function f the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivative f(epsilon)''(x0; d, d) when epsilon --> 0+ and both values are the same. The aim of the present work is to show the converse of this result. It will be shown that upper and lower limits of the approximate second-order directional derivative are equal to the second-order upper and lower Dini derivatives, respectively. Consequently the existence of the limit of the approximate second-order directional derivative and that of second-order Dini derivative are equivalent.