Entire bivariate functions of unbounded index in each direction - Bandura, A.I.; Skaskiv, O.B. (Нелінійні коливання, 2018)
We investigate a class of entire functions f(z₁, z₂) with property ∀b= (b₁, b₂) ∈ C² \ {0} ∀ z⁰₁, z⁰₂ ∈ C, the function f(z⁰₁ + tb₁, z⁰₂ + tb₂), as a function of one variable t ∈ C, has a bounded index but the function f(z₁, z₂) has an unbounded index in every direction b. In particular, we prove that, for an arbitrary even entire function f(t) that has an infinite sequence of complex zeros, the corresponding function f(√(z₁z₂)) has an unbounded index in every direction b. It improves our similar result [Bandura A. I. A class of entire functions of unbounded index in each direction // Mat. Stud. – 2015. – 44, № 1. – P. 107 – 112] proved for even entire functions f(t) with complex zeros ck such that c²k ∈ R.