A property of Peano derivatives in several variables

Fejzić, Hajrudin; Weil, Clifford

doi:10.1090/S0002-9939-2013-11529-7

A property of Peano derivatives in several variables
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by Hajrudin Fejzić and Clifford E. Weil PDF

Proc. Amer. Math. Soc. 141 (2013), 2411-2417 Request permission

Abstract:

Let $f$ be a function of several variables that is $n$ times Peano differentiable. Andreas Fischer proved that if there is a number $M$ such that $f_{\boldsymbol {\alpha } } \ge M$ or $f_{\boldsymbol {\alpha } } \le M$ for each $\boldsymbol {\alpha }$, with $\left | \boldsymbol {\alpha } \right | = n$, then $f$ is $n$ times differentiable in the usual sense. Here that result is improved to permit the type of one-sided boundedness to depend on $\boldsymbol {\alpha }$.

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