[7] **viXra:1908.0542 [pdf]**
*submitted on 2019-08-26 13:41:43*

**Authors:** Bulat Khabibullin, Enzhe Menshikova

**Comments:** 4 Pages.

We discuss the existence of an envelope of a function from a certain subclass of function space. Here we restrict ourselves to considering the model space of functions locally integrable with respect to the Lebesgue measure in a domain from the finite dimensional Euclidean space

**Category:** Functions and Analysis

[6] **viXra:1908.0511 [pdf]**
*submitted on 2019-08-25 14:34:25*

**Authors:** Bulat N. Khabibullin, Enzhe Menshikova

**Comments:** 5 Pages.

Let u and M are two non-trivial subharmonic functions in a domain D in the complex plane. We investigate two related but different problems. The first is to find the conditions on the Riesz measures of functions u and M respectively under which there exists a non-trivial subharmonic function h on D such that u+h< M. The second is the same question, but for a harmonic function h on D.
The answers to these questions are given in terms of the special affine balayage of measures introduced in our recent previous works. Applications of this technique concern the description of distribution of zeros for holomorphic functions f on the domain D satisfying the restriction |f|< exp M.

**Category:** Functions and Analysis

[5] **viXra:1908.0436 [pdf]**
*submitted on 2019-08-22 02:32:21*

**Authors:** Bulat N. Khabibullin, Enzhe Menshikova

**Comments:** 19 Pages.

We investigate some properties of balayage of measures and their potentials on domains or open sets in finite-dimensional Euclidean space. Main results are Duality Theorems for potentials of balayage of measures, for Arens-Singer and Jensen measures and potentials, and also a new extended and generalized variant of Poisson-Jensen formula for balayage of measure and their potentials.

**Category:** Functions and Analysis

[4] **viXra:1908.0434 [pdf]**
*replaced on 2019-09-01 13:45:41*

**Authors:** Kouider Mohammed Ridha

**Comments:** 6 Pages.

The most basic problem in numerical analysis (methods) is the root finding problem. In this paper we are interesting in new numerical method which based on the Modified Bisection Method(MBM) referred to by Tanakan superficially and he didn't know her as a numerical method for finding the roots of a function. Hence in this study we define a new numerical method base on MBM with error bound and the number iterations necessary. Finally we present our new MBM for multi-roots with the R software.

**Category:** Functions and Analysis

[3] **viXra:1908.0413 [pdf]**
*submitted on 2019-08-19 11:11:26*

**Authors:** Atabey Mahmudov

**Comments:** 3 Pages.

In this article, we introduce Cauchy's integral formula and proving by using analiticity of function inside of disk

**Category:** Functions and Analysis

[2] **viXra:1908.0296 [pdf]**
*submitted on 2019-08-15 06:47:48*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove an explicit bound for the isotropic constant in the symmetric case.

**Category:** Functions and Analysis

[1] **viXra:1908.0061 [pdf]**
*submitted on 2019-08-03 16:21:46*

**Authors:** Federico Espil

**Comments:** 2 Pages.

Shortly we can derive the Cauchy's residue theorem (its general form) just by integration of a Taylor Series "without" making any radius go to zero,even without the limit circumference idea take place. The Espil's theorem it's a short proof of the Cauchy's generalized residue theorem

**Category:** Functions and Analysis