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Lemma 2.3 .
Let f : ℝ + → ℝ + f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+} lim + ∞ f = 0 \lim_{+\infty}f=0 α , B > 0 \alpha,B>0 f f
H ( α , B ) t f ( s + t ) ≤ B [ f ( s ) ] 1 + α , ∀ s > 0 , ∀ 0 ≤ t ≤ 1 . H(\alpha,B)\hskip 28.45274pttf(s+t)\leq B[f(s)]^{1+\alpha},\;\forall s>0,\,\forall 0\leq t\leq 1.
Then there exists S ∞ = S ∞ ( α , B ) ∈ ℝ + S_{\infty}=S_{\infty}(\alpha,B)\in\mathbb{R}^{+} f ( s ) = 0 f(s)=0 s ≥ S ∞ s\geq S_{\infty}