The proofs of the extreme value theorem, the mean value theorem and the inverse function theorem for analytic functions on the Levi-Civita field will be presented. After reviewing convergence criteria for power series [15], we review their analytical properties [18, 20]. Then we derive necessary and sufficient conditions for the existence of relative extrema for analytic functions and use that ...

‎The subject of this paper is the solution of the Fredholm integral equation with Toeplitz, Hankel and the Toeplitz plus Hankel kernel. The mean value theorem for integrals is applied and then extended for solving high dimensional problems and finally, some example and graph of error function are presented to show the ability and simplicity of the ‎method.

In this paper, a new version of mean value theorem for interval-valued functions on time scales is established. Meantime, some basic concepts and results associated with semigroups operators are presented. As an application operators, under certain conditions, we consider the initial problem differential equations scales. Finally, two issues worthy further discussion

We show that the nth derivative of Riemann zeta function, when summed over non-trivial zeros zeta, is real and positive/negative in mean for n odd/even, respectively. this by giving a full asymptotic expansion these sums.