On the Possibility of Strengthening the Lieb–Thirring Inequality

In 1976, Lieb and Thirring obtained an upper bound for the square of the norm on $L^2(\mathbb R^2)$ of the sum of the squares of functions from finite orthonormal systems via the sum of the squares of the norms of their gradients. Later, a series of Lieb–Thirring inequalities for orthonormal systems was established by many authors. In the present paper, using the standard theory of functions, we prove Lieb–Thirring inequalities, which have applications in the theory of partial differential equations.