  In this study, Firstly, we will write two new convex functions for
$-1<n-\alpha \leq 1\ $and two new lemmas. Then we will find the relevance of
the two new lemmas to Caputo-left-sided derivatives under additional conditions
and draw conclusions about them. Secondly,We will give examples that support
it. In what follows, we will rewrite the Hermite Hadamard integral inequality
based on this new definitions. Furthermore, we will write an elementary
inequality valid for value $-1<n-\alpha \leq 1.$
