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We also show that both of these functions are of order two. These two functions will then be treated in more detail later in Chapters 6 and 7, respectively. Some other examples of tropical entire functions of arbitrary order are also presented as a formal ultra-discretization of complex entire functions given by q-series expansions. 1 Definitions and basic results Our key problem in this chapter is now to find relations between the order of growth, resp. 1) of a tropical entire function f (x) on the real line R corresponding to similar relations for their infinite power series expansions of entire functions on the complex plane C.

21) that have no contribution to the component of the graph of f (x). 21) is in the compact form if f (x) and any tropical entire function g(x) obtained as f (x) replaced its term cr∗ ⊗ x⊗jr∗ (r∗ ∈ Z≥0 ) by α ⊗ x⊗jr∗ with α < cr∗ are different functions. Intuitively, the maximal (and standard) representation of a tropical entire function f (x) is the compact form when all the terms contribute to the graph of f (x). 21) if the line jr x + cr remains below of the graph of f (x) except possible for one point where it touches the graph.

Hence, log (−ck )/sk log(−ck ) ≥ 1 + o(1) ρ+ε as k → ∞, since −ck → +∞. This gives lim inf k→∞ log (−ck )/sk log(−ck ) ≥ 1 ρ+ε or lim sup k→∞ log(−ck ) log (−ck )/sk ≤ρ+ε for any ε > 0, so that we have lim sup k→∞ log(−ck ) log (−ck )/sk ≤ lim sup r→∞ log M (r, f ) . log r To complete the proof of the lemma, assume now that the order ρ of f satisfies 1 < ρ < +∞, and consider the two limit superiors lim sup k→∞ −ck (−ck )/sk ρ and lim sup r→∞ M (r, f ) . rρ Moreover, suppose that the second one is finite and has the value λ, and thus for any ε > 0 there exists r0 > 0 such that ck + sk r ≤ M (r, f ) ≤ (λ + ε)rρ for any r ≥ r0 and for all k ∈ Z≥0 .

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A bivariate extension of Bleimann-Butzer-Hahn operator by Khan R.A.

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