Functions preserving general monotone sequences
D Torres-Latorre - Analysis Mathematica, 2021 - Springer
Analysis Mathematica, 2021•Springer
In this paper we characterize the functions that map the positive general monotone
sequences to themselves, ie, we obtain a necessary and a sufficient condition for the
functions φ to satisfy ∑ k= n^ 2n\left| a_ k+ 1-a_k\right| ≤ C a_n ⇒ ∑ k= n^ 2n\left| ϕ\left (a_
k+ 1\right)\right| ≤ C^ ′ ϕ\left (a_n\right).∑ k= n 2 n| ak+ 1− ak|≤ C an⇒∑ k= n 2 n| ϕ (ak+
1)|≤ C′ ϕ (an). for all\left {a_n\right\} _ n= 1^ ∞ ⊂ R^+ an n= 1∞⊂ ℝ+.
sequences to themselves, ie, we obtain a necessary and a sufficient condition for the
functions φ to satisfy ∑ k= n^ 2n\left| a_ k+ 1-a_k\right| ≤ C a_n ⇒ ∑ k= n^ 2n\left| ϕ\left (a_
k+ 1\right)\right| ≤ C^ ′ ϕ\left (a_n\right).∑ k= n 2 n| ak+ 1− ak|≤ C an⇒∑ k= n 2 n| ϕ (ak+
1)|≤ C′ ϕ (an). for all\left {a_n\right\} _ n= 1^ ∞ ⊂ R^+ an n= 1∞⊂ ℝ+.
In this paper we characterize the functions that map the positive general monotone sequences to themselves, ie, we obtain a necessary and a sufficient condition for the functions φ to satisfy ∑ k= n^ 2n\left| a_ k+ 1-a_k\right| ≤ C a_n ⇒ ∑ k= n^ 2n\left| ϕ\left (a_ k+ 1\right)\right| ≤ C^ ′ ϕ\left (a_n\right).∑ k= n 2 n| ak+ 1− ak|≤ C an⇒∑ k= n 2 n| ϕ (ak+ 1)|≤ C′ ϕ (an). for all\left {a_n\right\} _ n= 1^ ∞ ⊂ R^+ an n= 1∞⊂ ℝ+.
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