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It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]