Abstract.: We proceed with our study of increasing self-described sequences F, beginning with 1 and defined by a functional equation % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadA % eadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGGOaGaamyBaiaacMca % caGG8bGaeyypa0ZaaebeaeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaO % Gaaiikaiaad2gacaGGPaWaaWbaaSqabeaacaWGHbWaaSbaaWqaaiaa % dMgaaeqaaaaakiaacIcacaaIXaGaey4kaSIaam4BaiaacIcacaaIXa % GaaiykaiaacMcaaSqaaiaaicdacqGHKjYOcaWGPbGaeyizImQaam4A % aaqab0Gaey4dIunakiaaysW7caGGOaGaamyyamaaBaaaleaacaWGPb % aabeaakiabgwMiZkaaicdacaaMe8Uaaeyyaiaab6gacaqGKbGaaGjb % VlaadAeadaWgaaWcbaGaamyAaaqabaGccaaMe8Uaaeizaiaabwgaca % qGUbGaae4BaiaabshacaqGPbGaaeOBaiaabEgacaaMe8UaamOraiab % lIHiVjabl+UimjablIHiVjaadAeacaGGPaGaaiOlaaaa!729F! $$|F^{ - 1} (m)| = \prod\nolimits_{0 \leq i \leq k} {F_i (m)^{a_i } (1 + o(1))} \;(a_i \geq 0\;{\text{and}}\;F_i \;{\text{denoting}}\;F \circ \cdots \circ F).$$ In [1] we exhibited the simple solution f′ (t)=Ctβ, for some β ∈(0,1), of the associated functional-differential equation % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafa % GaaiikaiaadshacaGGPaGaeyypa0ZaaebeaeaacaWGMbWaaSbaaSqa % aiaadMgaaeqaaOGaaiikaiaadshacaGGPaWaaWbaaSqabeaacqGHsi % slcaWGHbWaaSbaaWqaaiaadMgaaeqaaaaaaSqaaiaaicdacqGHKjYO % caWGPbGaeyizImQaam4Aaaqab0Gaey4dIunakiaacYcaaaa!4A46! $$f'(t) = \prod\nolimits_{0 \leq i \leq k} {f_i (t)^{ - a_i } } ,$$ and we proved that provided β<2/(2+d(Γ)), where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaacI % cacqqHtoWrcaGGPaGaaiOoaiabg2da9iaadggadaWgaaWcbaGaaGym % aaqabaGccqGHRaWkcqWIVlctcqGHRaWkcaWGHbWaaSbaaSqaaiaadU % gaaeqaaOGaaiilaaaa!43A0! $$d(\Gamma ): = a_1 + \cdots + a_k ,$$ we have the asymtotic equivalence F(m)~ Cmβ. In the present paper we show that this last result is optimal, in the sense that the self-described sequence defined by |F−1(m)|=F(m)2, that is $$ 1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,7, \ldots , $$ for which the boundary case β=2/(2+d(Γ))(=1/2) holds, does not satisfy F(m) ~ Cmβ. We also show that the m-th term F(m) of a sequence F for which the boundary case holds is nevertheless of asymptotic order mβ. Then we investigate the behaviour of self-described sequences F when β lies beyond the boundary case. In [1] we established the estimates % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa % aaleqabaGaeqOSdiMaeyOeI0IaeqyTdugaaOGaeSOAI0JaamOraiaa % cIcacaWGTbGaaiykaiablQMi9iaad2gadaahaaWcbeqaaiabek7aIj % abgUcaRiabew7aLbaakiaacIcacaGGQaGaaiykaaaa!4870! $$m^{\beta - \varepsilon } \ll F(m) \ll m^{\beta + \varepsilon } (*)$$ when β is the unique fixed point of a certain associated function. We were only able to prove in general that the latter holds when β does not lie beyond the boundary case, however. In the present paper we prove that whenever % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey % izIm6aaSGbaeaacaaIXaaabaWaaOaaaeaacaaIXaGaey4kaSIaamiz % aiaacIcacqqHtoWrcaGGPaaaleqaaaaakiaacYcaaaa!4030! $$\beta \leq {1 \mathord{\left/ {\vphantom {1 {\sqrt {1 + d(\Gamma )} }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 + d(\Gamma )} }},$$ β is the unique fixed point of this function, and in addition we obtain estimates more precise than (*). This applies for instance to the sequence defined by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaadA % eadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGGOaGaamyBaiaacMca % caGG8bGaeyypa0JaaiikaiaadAeacqWIyiYBcaWGgbGaaiykaiaacI % cacaWGTbGaaiykaiaacYcaaaa!450C! $$|F^{ - 1} (m)| = (F \circ F)(m),$$ that is $$ 1,2,2,3,3,4,4,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,10,11, \ldots .. $$

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{{Explor lien
   |wiki=    Wicri/Lorraine
   |area=    InforLorV4
   |flux=    Main
   |étape=   Merge
   |type=    RBID
   |clé=     ISTEX:7B775BB44920FC25CCF8CD1EF254F33240D984F1
   |texte=   Increasing Self-Described Sequences
}}