Generalized Mean-Value Theorem for an Analytic Function and an Algorithm for the Evaluation of the Function of Mean Values

2017 ◽  
Vol 69 (6) ◽  
pp. 892-915
Author(s):  
A. M. Samoilenko
Keyword(s):  
Analytic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Values ◽  
Generalized Mean

scholarly journals Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

Exploring students’ proof comprehension of the Cauchy Generalized Mean Value Theorem

2019 ◽  
Vol 39 (3) ◽  
pp. 213-235
Author(s):  
Fahimeh Kolahdouz ◽  
Farzad Radmehr ◽  
Hassan Alamolhodaei
Keyword(s):  
Undergraduate Students ◽  
Mean Value ◽  
Mean Value Theorem ◽  
First Year ◽  
Test Design ◽  
Test Results ◽  
Mathematical Proofs ◽  
Proof Comprehension ◽  
Generalized Mean ◽  
First Year University

Abstract Undergraduate students majoring in mathematics often face difficulties in comprehending mathematical proofs. Inspired by a number of studies related to students’ proof comprehension, and Mejia-Ramos et al.’s study in particular, a test was designed in relation to the proof comprehension of the Cauchy Generalized Mean Value Theorem (CGMVT). The test mainly focused on (a.) investigating students’ understanding of relations between the statements within the CGMVT proof and (b.) the relations between the CGMVT and other theorems. Thirty-five first-year university students voluntarily participated in this study. In addition, 10 of these students were subsequently interviewed to seek their opinion about the test. Test results indicated that most of the students lacked an understanding of the relations between the mathematical statements within the CGMVT proof, and the relations between the CGMVT and other theorems. The results of interviews showed that this type of assessment was new to students and helped them to improve their insights into mathematical proofs. The findings suggested such a test design could be used more frequently in assessments to aid instructors’ understanding of students’ proof comprehension and to teach students how mathematical proofs should be learned.

Mean values and classes of harmonic functions

1984 ◽  
Vol 96 (3) ◽  
pp. 501-505 ◽  
Author(s):  
Thomas Ramsey ◽  
Yitzhak Weit
Keyword(s):  
Unit Circle ◽  
Harmonic Functions ◽  
Borel Measure ◽  
Mean Value ◽  
Finite Complex ◽  
Mean Values ◽  
Complex Borel Measure ◽  
Generalized Mean ◽  
Image Position

Let μ be a finite complex Borel measure supported on the unit circle.In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form.and for all ξ ∈ , | ξ | = R for some fixed R > 0.

scholarly journals On some mean value theorems of the differential calculus

1971 ◽  
Vol 5 (2) ◽  
pp. 227-238 ◽  
Author(s):  
J.B. Diaz ◽  
R. Výborný
Keyword(s):  
Differential Calculus ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Value Theorems ◽  
Partial Converse ◽  
Generalized Mean ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Special Case

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

scholarly journals A local mean-value theorem for analytic functions with smooth boundary values

1974 ◽  
Vol 15 (1) ◽  
pp. 27-29 ◽  
Author(s):  
W. P. Novinger
Keyword(s):  
Analytic Function ◽  
Direct Consequence ◽  
Mean Value ◽  
Open Mapping ◽  
Mean Value Theorem ◽  
Open Mapping Theorem ◽  
Easy Consequence ◽  
Open Set ◽  
One To One ◽  
Image Position

Let f be an analytic function on a connected open set Ώ in the complex plane. Then, for given a, b ∈ Ώ, the equationneed not have a solution z ∈ Ώ. As a matter of fact, this would happen with each locally one-to-one analytic function which is not one-to-one on Ώ. But if we fix a a ∈ Ώ, then, for all b sufficiently close to a, (1) is solvable for z. This is an easy consequence of the Open Mapping Theorem applied to f'. For, assuming that f' is non-constant (otherwise, (1) holds for all a, b, z ∈Ώ), the Open Mapping Theorem tells us that f'(Ώ), the image under f' of Ώ, is an open neighbourhood of f'(a); so it is a direct consequence of the definition of f'(a) that there exists δ > 0 such that 0 < |b – a| < δ implies (f(b) – f(a))/(b – a) ∈f'(Ώ). A stronger statement has been obtained by J. M. Robertson [1, p. 329], who has shown thatand, if f''(a) ≠ 0, then.

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