Definition 6 The second-order divided difference of a function f : [ a, b ] → R at mutually distinct points y 0, y 1, y 2 ∈ [ a, b ] is defined recursively by (23).
Let (D_{1}={f_{t}:tin I}) be a class of functions such that the function (tmapsto[ z_{0},z_{1},ldots,z_{n};f_{t}]) is n-exponentially convex in the J-sense on I for any (n+1) mutually distinct points (z_{0},z_{1},ldots,z_{n}in[ a,b]).
Theorem 3.7 Let Ω = { f s : s ∈ I ⊆ R } be a family of differentiable functions defined on [ a, b ] such that the function s ↦ [ y 0, y 1, y 2 ; f s ] is n-exponentially convex in the Jensen sense on I for every three mutually distinct points y 0, y 1, y 2 ∈ [ a, b ].
Corollary 3.8 Let Ω = { f s : s ∈ I ⊆ R } be a family of differentiable functions defined on [ a, b ] such that the function s ↦ [ y 0, y 1, y 2 ; f s ] is exponentially convex in the Jensen sense on I for every three mutually distinct points y 0, y 1, y 2 ∈ [ a, b ].
Let ({D}_{2}={f_{t}:tin I}) be a class of functions such that the function (tmapsto [ z_{0},z_{1},ldots,z_{n};f_{t}]) is exponentially convex in the J-sense on I for any (n+1) mutually distinct points (z_{0},z_{1},ldots,z_{n}in [ a,b]).
Take mutually distinct points (x_{n}in Omega ) and define (h_{n}in C_{0} overline{B(x_{n},varepsilon_{n})})) by h_{n}(x)= textstylebegin{cases} 0, & vert x-x_{n}vert geq varepsilon_{n}, varepsilon_{n}-vert x-x_{n}vert, & vert x-x_{n}vert < varepsilon_{n}, end{cases} for (varepsilon_{n}>0) with (overline{B(x_{n},varepsilon_{n})} subset Omega ).
Suppose and where the complex numbers are mutually distinct with, and forms a complex system of mutually disjoint projections on.
for any θ 0,θ 1,θ 2>0 and any mutually distinct x 0,x 1,x 2∈V.
Verify if the fingerprints f(n 1),…,f(f α ) are mutually distinct.
Conditions under which every eventually positive solution belongs to one of three mutually distinct sets are derived.
while the cases where are mutually distinct and will be considered in a subsequent paper (for the important reason that quite distinct techniques are needed).
