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We study ternary monomial functions of the form $f(x)=\Tr_n(ax^d)$, where$x\in \Ff_{3^n}$ and $\Tr_n: \Ff_{3^n}\to \Ff_3$ is the absolute tracefunction. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gausssums, and certain ternary weight inequalities, we show that certain ternarymonomial functions arising from \cite{hk1} are weakly regular bent, settling aconjecture of Helleseth and Kholosha \cite{hk1}. We also prove that theCoulter-Matthews bent functions are weakly regular.

Proofs of two conjectures on ternary weakly regular bent functions