Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

In this paper, we investigate the universal inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in an n-dimensional complete noncompact simply connected Riemannian manifold with its sectional curvature satisfying certain pinching conditions, in a Gaussian shrinking soliton, and in a cigar metric measure space, respectively. By using some analytic inequalities and geometric inequalities, we establish some new universal inequalities which are different from those already present in the literature, such as Yanli Li and Feng Du's [Arch Math (Basel) 109(6):591-598, 2017], Feng Du et al.'s [Z Angew Math Phys 66(3):703-726, (2015)] and Xinyang Li, Xin Xiong and Lingzhong Zeng's [J Geom Phys 145:103472, (2019)].