Geodesic
Applications of a rather general mean value theorem for integrals
Viorel Vîjîitu1 ; Viorel Vîjîitu
1Laboratoire P. Painlevé, Université de Lille, F-59655 Villeneuve d'Ascq Cedex, Lille, France
Acta mathematica Universitatis Comenianae, Tome 94 (2025) no. 2, pp. 59-64 
Viorel Vîjîitu; Viorel Vîjîitu. Applications of a rather general mean value theorem for integrals. Acta mathematica Universitatis Comenianae, Tome 94 (2025) no. 2, pp. 59-64. http://geodesic.mathdoc.fr/item/AMUC_2025_94_2_a1/
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     author = {Viorel V{\^\i}j{\^\i}itu and Viorel V{\^\i}j{\^\i}itu},
     title = { Applications of a rather general mean value theorem for integrals},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {59--64},
     year = {2025},
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     url = {http://geodesic.mathdoc.fr/item/AMUC_2025_94_2_a1/}
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Voir la notice de l'article provenant de la source Comenius University

In this paper, we present mean value results for Volterra's operator and for integral of Flett's type. These are based on the following mean value theorem for integrals, due to the author:Let $a$ and $b$ be real numbers such that $a, and let $\mathfrak{f}: [a,b] \longrightarrow \mathbb{R}$ be a gauge integrable function with $\int_a^b \mathfrak{f}=0$. Then, for every ascending function $\mathfrak{z}: [a,b] \longrightarrow [0,\infty)$ which is nonconstant on $(a,b)$, there exists a point $c \in (a,b)$ that satisfies$\int_a^c \mathfrak{z} \mathfrak{f}=0.$We also generalize several recently published problems to their natural setting.