| Abstract: |
| Let $K$ be a convex compact subset of $R^p, p\geq 1$, having nonempty interior. Starting with a suitable positive linear projection $T$ defined on $C(K)$, Altomare and Rasa defined in [1] the weakly $T$-convex functions. Using $T$, a $C_0$-semigroup of operators on C(K) was constructed and the generalized $A$-subharmonic functions were defined, where $A$ is the infinitesimal generator of the semigroup. It was proved that if a function is weakly $T$-convex, then it is generalized $A$-subharmonic. The authors of [1] conjectured that the converse is also true, but as far as we know this is still an open problem. We present some results related to the conjecture. Namely, starting with the conjecture, we prove that a suitable stronger hypothesis entails a stronger conclusion.
[1] F. Altomare, I. Rasa, Feller semigroups, Bernstein type operators and generalized convexity associated with positive projections, New Developments in Approximation Theory, Internat. Ser. Numer. Math. vol.132, Birkhauser, Basel, 1999, 9-32. |
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