ScalingStacks

Lemma 4 [0292]

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Lemma 4

. Let a:(−∞,0]→[0,1]a:(-\infty,0]\rightarrow[0,1], be a monotone non decreasing function such that for some B>0B>0, δ>0\delta>0 the inequality

t​a​(s)≤B​a​(s+t)1+δt\,a(s)\leq B\,a(s+t)^{1+\delta}

holds for all s≤0,t∈[0,1],s+t≤0s\leq 0,\,t\in[0,1],\,s+t\leq 0. Then for all S<0S<0 such that a⁡(S)>0a(S)>0 and all D∈[0,1],S+D≤0D\in[0,1],\,S+D\leq 0 we have the estimate

D≤e⁡(3+2/δ)​B​a​(S+D)δ.D\leq e(3+2/\delta)B\,a(S+D)^{\delta}\,.

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