Using z f ′ ( z ) instead of f in Corollary 2.8, we obtain the order of convexity for a function satisfying (2.22), so we have | z f ″ ( z ) + f ′ ( z ) − 1 | < λ | f ′ ( z ) | ⇒ ( 1 + z f ″ ( z ) f ′ ( z ) ) > α, where α is described in (2.30), which means that f ∈ K α.
To prove the last inequality in (6), we will use Corollary 2 and 5 instead of Proposition 3 and Corollary 4. The same argument as showing the last inequality in (3) gives the last inequality in (6).
As already mentioned in Section 1.D, any countable elementary amenable group embeds in a finitely generated elementary amenable group; the same hold for "amenable" instead of "elementary amenable" [128, Corollary 1.3].
Note that if we use the functions (F_{1,-alpha }), instead of their normalisations, in the proof of Corollary 4.3 we can cover the case that ({underline{u}}) is an admissible strict subsolution of (1.1)–(1.2), without having to modify B. Using the approximations (F_{k,-alpha }) in the proof of Corollary 4.2, we can similarly adjust the proof of Corollary 4.2.
We utilize Corollary 3.5 instead of Theorem 2.7 in the proof of Theorem 2.12.
Note that (1) If X = D, then, by Lemma 4.3, in the condition (2) of Theorem 4.1, we can put G(Γ(A)) ⊆ H(A) for all A ∈ ⟨D⟩ and so T(Γ(A)) ⊆ H(A) for all A ∈ ⟨D⟩ instead of the condition (1) in Corollary 4.2.
Corollary 2.2 Assume the conditions (25) and (26) instead of continuity of F in Corollary 2.1, then the conclusion of Corollary 2.1 holds.
On taking Φ instead of Ω and compatibility instead of O-compatibility in Theorem 4 (with assumption (e) only), we obtain a sharpened version of Corollary 12.
In fact on taking Φ instead of Ω in Corollary 2 (with assumption (e2)′ only), we obtain correct form of Corollary 11.
