Some regularity properties of the functions obtained from some algebraic properties

Marinescu, Dan Ştefan and Monea, Mihai

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creative_2020_29_2_161_164

DOI: https://doi.org/10.37193/CMI.2020.02.07

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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 $\left( a,b\right)$ .