### A maximal inequality of non-negative submartingale.

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In this paper we establish some new nonlinear difference inequalities. We also present an application of one inequality to certain nonlinear sum-difference equation.

We characterize homogeneous real hypersurfaces of types (A₀), (A₁) and (B) in a complex projective space or a complex hyperbolic space.

We study rotation surfaces in the three-dimensional pseudo-Galilean space G₃¹ such that the Gauss map G satisfies the condition L₁G = f(G + C) for a smooth function f and a constant vector C, where L₁ is the Cheng-Yau operator.

The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map G of a circular cone, one has ΔG = f(G+C), where Δ is the Laplacian operator, f is a non-zero function and C is a constant vector. We prove that circular cones are characterized by being...

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