We prove, among other results, that if $ f $ is an analytic function on the open unit disc in the complex plane belonging to the Bloch space $ {\mathcal {B}} $, then the asymptotic relation holds
$ {\rm dist}(f, {\mathcal {B}}_0)\asymp\limsup\limits_{|z|\to1-0}(1-|z|^2)^l|f^{(l)}(z)|, $
for each $ l\in {\mathbb N}, $ where
$ {\rm dist}(f, {\mathcal {B}}_0) = \inf\limits_{g\in{\mathcal {B}}_0}\|f-g\|_{\mathcal {B}} $
and $ {\mathcal {B}}_0 $ is the little Bloch space, extending a result in the literature concerning the case $ l = 1 $. In passing, we also obtain the following asymptotic relation
$ {\rm dist}(f, {\mathcal {B}}_0)\asymp\limsup\limits_{r\to1-0}\|f-f_r\|_{\mathcal {B}}. $
Citation: Stevo Stević. Note on some asymptotic relations for Bloch functions and the distance from a Bloch function to the little Bloch space[J]. AIMS Mathematics, 2026, 11(3): 7687-7694. doi: 10.3934/math.2026316
We prove, among other results, that if $ f $ is an analytic function on the open unit disc in the complex plane belonging to the Bloch space $ {\mathcal {B}} $, then the asymptotic relation holds
$ {\rm dist}(f, {\mathcal {B}}_0)\asymp\limsup\limits_{|z|\to1-0}(1-|z|^2)^l|f^{(l)}(z)|, $
for each $ l\in {\mathbb N}, $ where
$ {\rm dist}(f, {\mathcal {B}}_0) = \inf\limits_{g\in{\mathcal {B}}_0}\|f-g\|_{\mathcal {B}} $
and $ {\mathcal {B}}_0 $ is the little Bloch space, extending a result in the literature concerning the case $ l = 1 $. In passing, we also obtain the following asymptotic relation
$ {\rm dist}(f, {\mathcal {B}}_0)\asymp\limsup\limits_{r\to1-0}\|f-f_r\|_{\mathcal {B}}. $
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Stevo Stević. Note on some asymptotic relations for Bloch functions and the distance from a Bloch function to the little Bloch space[J]. AIMS Mathematics, 2026, 11(3): 7687-7694. doi: 10.3934/math.2026316
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