Schur-type inequality for solitonic hypersurfaces in $(k,\mu )$-contact metric manifolds

In this article, we derive a Schur-type Inequality in terms of the gradient $r$-Almost Newton-Ricci-Yamabe soliton in $(k,\mu )$-contact metric manifolds. We discuss the triviality for the compact gradient $r$-Almost Newton-Ricci-Yamabe soliton in $(k,\mu )$-Contact metric manifolds. In the end, we deduce a Schur-type inequality for the gradient $r$-Almost Newton-Yamabe soliton in $(k,\mu )$-contact metric manifolds, static Riemannian manifolds, and normal homogeneous compact Riemannian manifolds coupled with a projected Casimir operator.