Čebyšev-Grüss inequalities for \(\alpha\)-partial derivatives (Q2211164)

Given a function \(f:[a_1,b_1]\times\dots \times[a_n,b_n]\rightarrow \mathbb{R}\) and \(\alpha \in (0,1]\), the author defines the \(\alpha\)-multivariate partial derivative by \[ \frac{\partial}{\partial x_i}(f)^{a_i}_\alpha (x_1, \dots, x_n):=\lim_{\epsilon\rightarrow 0} \frac{f(x_1, \dots, x_{i-1},x_i+\epsilon x_i^{1-\alpha}, x_{i+1},\dots,x_n)-f(x_1, \dots, x_n)}{\epsilon}, \] and then he proves Lagrange's mean value theorem, Cauchy's mean value theorem and Rolle's theorem for functions with \(\alpha\)-multivariate partial derivatives. Next, the author introduces the \(\alpha\)-multivariate integral as follows: If \(\alpha \in (0,1], 0\le a_i <b_i \) for all \(i \in \{1, \dots, n\}\), and \(f:[a_1,b_1]\times\dots \times[a_n,b_n]\rightarrow\mathbb{R}\), then \begin{align*} &\int_{a_1}^{b_1} \cdots \int_{a_n}^{b_n} f(x_1, \dots, x_n)\, d_\alpha x_1 \dots d_\alpha x_n:= \\ &\quad\int_{a_1}^{b_1} \cdots \int_{a_n}^{b_n} (x_1 \dots x_n)^{\alpha -1}f(x_1, \dots, x_n)\,dx_1 \dots dx_n. \end{align*} Finally, these notions are applied to prove some Chebyshev-Grüss type inequalities for functions which are \(\alpha\)-multivariate differentiable.