Logarithmic source indices and their admissible parameter scales

On $(0,t_0)$ write $L=\log (1\mathrm{/}t)$. Direct differentiation gives ${\psi }^{\mathrm{\prime }}(t)=C\nu t^{-1}{L}^{-\nu -1}>0$ and ${\psi }^{\mathrm{\prime }\mathrm{\prime }}(t)=C\nu t^{-2}{L}^{-\nu -2}(\nu +1-L)<0$. At $t_0$ the derivative drops from a positive value to zero; above it the function is constant. Thus the derivative is nonincreasing on each smooth piece and has a downward jump, giving concavity on $(0,\mathrm{\infty })$; taking limits at zero extends the chord inequalities there. Continuity and monotonicity are immediate from the formula. In particular $H$ is finite because $0\le \psi \le \psi (t_0)$, and $H\ge 0$ by testing $t=0$.

Put $Q=\log q$ and take $q$ sufficiently large. For $0<t\le q^{-1}{Q}^{\nu +1}$, one has $\log (1\mathrm{/}t)\ge Q-(\nu +1)\log Q$, so $\psi (t)-qt\le C[Q-(\nu +1)\log Q{]}^{-\nu }$. For $t>q^{-1}{Q}^{\nu +1}$, one has $qt>Q^{\nu +1}>\psi (t_0)$, so the expression is negative. This bounds $H(q)$ above by $C{Q}^{-\nu }(1+o(1))$. Testing $t=q^{-1}{Q}^{-\nu -1}$ gives $\psi (t)-qt=C[Q+(\nu +1)\log Q{]}^{-\nu }-Q^{-\nu -1}=C{Q}^{-\nu }(1+o(1))$, proving the matching lower bound. Replacing $q$ by $cq$ changes $Q$ by the fixed constant $\log c$, hence $H(cq)\mathrm{/}H(q)\to 1$.

For the balanced scale, $\log (1\mathrm{/}\alpha )\sim p\log (1\mathrm{/}\delta )$, so step 2.1 gives both conjugate values in [F1] of order $[\log (1\mathrm{/}\delta ){]}^{-\nu }$. Moreover ${\delta }^p\mathrm{/}\alpha$ has this same order, and $\alpha H(1\mathrm{/}(2C_{\ast }\alpha ))=O({\delta }^p)$. Substitution in [F1] proves the balanced error and residual bounds.

If $\alpha \asymp {\delta }^k$ with $0<k<p$, step 2.1 again gives the two conjugate values of order $[\log (1\mathrm{/}\delta ){]}^{-\nu }$. Since ${\delta }^{p-k}[\log (1\mathrm{/}\delta ){]}^{\nu }\to 0$ (put $x=\log (1\mathrm{/}\delta )$ and compare the exponential $e^{(p-k)x}$ with its power-series term of integer degree greater than $\nu$), ${\delta }^p\mathrm{/}\alpha$ is bounded by the desired logarithmic order. Also ${\delta }^p=O({\delta }^k[\log (1\mathrm{/}\delta ){]}^{-\nu })$. Apply [F1] for the two asserted bounds. Finally, $e_{\delta }=\frac{1}{2}\mathrm{\parallel }{u}_{\delta }-u^{†}{\mathrm{\parallel }}^2$ gives the norm conclusion by taking square roots.