Applications of a rather general mean value theorem for integrals

Abstract

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 $a<b$, 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.