On connections between approximate second-order directional derivative and second-order Dini derivative for convex functions

For a real-valued convex functionf, the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivativef (x ₀;d, d) when 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.Dedicated to Professor N. Furukawa of Kyushu University for his 60th birthday.