The main result of this note is stated as follows.
Let a,b\in \mathbb{R}, a<b, and a function f:\left( a,b\right) \rightarrow \mathbb{R}. We consider u,v:\left( a,b\right) \rightarrow \mathbb{R} such that u\left( x\right) >0 and v\left( x\right) <0, for any x\in \left( a,b\right), and define g,h:\left( a,b\right) \rightarrow \mathbb{R} by g\left( x\right) =u\left( x\right) f\left( x\right) and h\left(x\right) =v\left( x\right) f\left( x\right), for any x\in \left( a,b\right). The main result we establish is stated as follows:

Theorem. Let n  a positiveinteger, n\geq 3.  If u,v are \left( n-1\right)-times differentiable on \left( a,b\right) and g,h are n-convex functions then f is \left(n-1\right)-times differentiable on \left( a,b\right).





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  Monea, Mihai ,  Marinescu, Dan Ştefan