In this paper, based on the classical Yano’s formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in complex space forms, which involves Ricci curvature direction \(J\mathbf {H}\) and norm of covariant differentiation second fundamental form h, where J is almost structure \(\mathbf mean vector field. Second analogously, Legendrian Sasakian forms with \(...

It is well known that the two-sample Student t test fails to maintain its significance level when the variances of treatment groups are unequal, and, at the same time, sample sizes are unequal. However, introductory textbooks in psychology and education often maintain that the test is robust to variance heterogeneity when sample sizes are equal. The present study discloses that, for a wide vari...