This article studies a pharmacokinetics problem, which is the mathematical modeling of a drug concentration variation in human blood, starting from the injection time. Theories and applications of fractional calculus are the main tools through which we establish main results. The psi-Caputo fractional derivative plays a substantial role in the study. We prove the existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. The application of the theoretical results on two data sets shows the following results. For the first data set, a psi-Caputo with the kernel ψ = x + 1 is the best approach as it yields a mean square error (MSE) of 0.04065 . The second best is the simple fractional method whose MSE is 0.05814 ; finally, the classical approach is in the third position with an MSE of 0.07299 . For the second data set, a psi-Caputo with the kernel ψ = x + 1 is the best approach as it yields an MSE of 0.03482 . The second best is the simple fractional method whose MSE is 0.04116 and, finally, the classical approach with an MSE of 0.048640 .

In this paper we study the existence of Lelong numbers of m−subharmonic currents of bidimension (p, p) on an open subset of C n , when m+p ≥ n. In the special case of m−subharmonic function ϕ, we give a relationship between the Lelong numbers of dd c ϕ and the mean values of ϕ on spheres or balls. As an application we study the integrability exponent of ϕ. We express the integrability exponent of ϕ in terms of volume of sub-level sets of ϕ and we give a link between this exponent and its Lelong number.2010 Mathematics Subject Classification. 32U25; 32U40; 32U05.

Presented by Jean-Pierre Demailly I dedicate this work to the martyrs of the Tunisian revolution, in particular to my colleague Hatem Bettaher. In this Note we study the existence of the Lelong-Demailly number of a negative plurisubharmonic current with respect to a positive plurisubharmonic function on an open subset of C n . Then we establish some estimates of the Lelong-Demailly numbers of positive or negative plurisubharmonic currents. © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Dans cette Note, on étudie l'existence du nombre de Lelong-Demailly d'un courant négatif plurisousharmonique relativement à une fonction positive plurisousharmonique sur un ouvert de C n puis on donne quelques estimations des nombres de Lelong-Demailly des courants positifs ou négatifs plurisousharmoniques. Version française abrégéeL'existence des nombres de Lelong des courants positifs a été résolu par P. Lelong dans les années 1950 pour le cas des courants fermés, puis ce résultat a été étendu par Skoda au cas des courants positifs plurisousharmoniques. En revanche, il existe des courants négatifs plurisousharmoniques qui n'admettent pas de nombres de Lelong, on peut voir par exemple que log(|z 2 | 2 )[z 1 = 0] est un exemple de courant négatif plurisousharmonique de bidimension (1, 1) sur la boule unité de C 2 qui n'admet pas de nombre de Lelong en 0.Le principal objectif de cette note est de traiter le problème de l'existence des nombres de Lelong généralisés introduits par Demailly [1,2] d'un courant négatif plurisousharmonique T de bidimension (p, p) sur un ouvert Ω de C n ; pour cela on note PSH(T , Ω) l'ensemble des fonctions ϕ positives plurisousharmoniques semi-exhaustives dont le logarithme log ϕ est plurisousharmonique sur Ω, et telles que le produit extérieur T ∧ (dd c ϕ) p soit bien défini. Le nombre de Lelong-Demailly de T relativement à un poids ϕ tel que ϕ ∈ PSH(T , Ω) est ν(T , ϕ) := lim r→0 + ν(T , ϕ, r), où ν(T , ϕ, .) est la fonction définie par ν(T , ϕ, r) := 1 r p {ϕ

In this paper we study the continuity of the Berezin transform on modified Bergman spaces and we establish a Lipschitz estimate in terms of the Bergman-Poincaré metric.

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This article studies a pharmacokinetics problem which is the mathematical modeling of a drug concentration in a human blood over time, starting from when the drug is administered. Results are obtained using fractional calculus theories. A fractional derivative known as Psi-Caputo plays a substantial role in the study. We proved existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. Application of the results on a data set showed that a psi-Caputo with the kernel ψ = x + 1 was the best approach as it leaded to mean square error of 0.04065. The second best was the simple fractional method whose error was 0.05814 and, finally the classical approach with an error of 0.07299.

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