We study the uniqueness question of transcendental entire functions sharing one finite nonzero value with their derivatives in some angular domains instead of the whole complex plane. The results in the present paper improve and extend the corresponding results from Chang-Fang~[2] and extend Theorem 3 from Zheng~[15]. An example is provided to show that the results in this paper, in a sense, are best possible.

By applying a comparison theorem on trajectory-harps, we give an estimate of volumes of trajectory-balls for Kähler magnetic fields from below under an assumption that sectional curvatures of the underlying Kähler manifold are bounded from above.

The main purpose of this article is to study the Caffarelli-Kohn-Nirenberg type inequalities (1.2) with $p=1$. We show that symmetry breaking of the best constants occurs provided that a parameter $|\gamma|$ is large enough. In the argument we effectively employ equivalence between the Caffarelli-Kohn-Nirenberg type inequalities with $p=1$ and the isoperimetric inequalities with weights.

We will see an alternative proof of the non-Archimedean Montel theorem for rational dynamics by using the non-Archimedean Green functions.