scholarly journals Inequalities for generalized divisor functions

2021 ◽  
Vol 27 (2) ◽  
pp. 41-48
Author(s):  
József Sándor ◽  
Keyword(s):  
Real Variable ◽  
The Real ◽  
Divisor Functions ◽  
Convexity Properties

We offer inequalities to $\sigma_a(n)$ as a function of the real variable $a$: Monotonicity and convexity properties to this and related functions are proved, too. Extensions and improvements of known results are provided.

scholarly journals Generalised sturm-liouville expansions in series of pairs of related functions

1927 ◽  
Vol 115 (770) ◽  
pp. 184-198 ◽  
Keyword(s):  
Boundary Conditions ◽  
Differentiable Function ◽  
Real Variable ◽  
Complex Parameter ◽  
The Real ◽  
The Earth ◽  
In Series ◽  
Sturm Liouville

The expansions here developed are required for the author’s discussion of "Meteorological Perturbations of Tides and Currents in an Unlimited Channel rotating with the Earth” ( v. supra , p. 170). Let η ( x ) be a real differentiable function of x defined in the range 0 ≼ x ≼ 1, and satisfying the condition η ( x ) > c > 0 for all such x . Let ϕ λ ( x ) and ψ λ ( x ) be functions of the real variable x and the complex parameter λ , defined in the above range by the equations d / dx [ η ( x ) dϕ λ ( x )/ dx ] + ( λ + iγ ) ϕ λ ( x ) = -1, d / dx [ η ( x ) dψ λ ( x )/ dx ] + ( λ + iγ ) ψ λ ( x ) = -1 (1) together with the boundary conditions ϕ' λ (0) = 0, ψ' λ (0) = 0, ϕ' λ (1) = 0, ψ λ (1) = 0, (2) γ being a prescribed constant.

On the Closure of and Integral Functions

1935 ◽  
Vol 31 (3) ◽  
pp. 335-346 ◽  
Author(s):  
Norman Levinson
Keyword(s):  
Real Variable ◽  
The Real ◽  
Almost Everywhere ◽  
Complex Valued

1. A set of functions {øn (x)} is said to be closed L over an interval (a, b) if for an f (x) belonging to Limplies that f(x) = 0 almost everywhere. Here f(x) is a complex valued function of the real variable x.

scholarly journals Fourier's series and analytic functions

1922 ◽  
Vol 101 (709) ◽  
pp. 124-133 ◽  
Keyword(s):  
Complex Plane ◽  
Analytic Functions ◽  
Real Variable ◽  
The Real ◽  
Real Value

1.1. In the theorems which follow we are concerned with functions f ( x ) real for real x and integrable in the sense of Lebesgue. We do not, however, remain in the field of the real variable, for we suppose, in 4 et seq ., that f ( x ), or a function associated with f ( x ), is analytic, or, at any rate, harmonic, in a region of the complex plane associated with the particular real value of x considered. The Fourier’s series considered are those associated with the interval (0, 2π). If a is a point of the interval, we write ϕ(u) = ½ { f ( a + u ) + f ( a - u ) - 2 s } (0 < a < 2π), (1.11) ϕ(u) = ½ { f ( u ) + f (2π - u ) - 2 s } ( a = 0, a = 2π), (1.12) where s is a constant.

On the derivation of discontinuous functions

1939 ◽  
Vol 35 (3) ◽  
pp. 373-381
Author(s):  
D. R. Dickinson
Keyword(s):  
Real Function ◽  
Limit Point ◽  
Real Variable ◽  
Positive Direction ◽  
The Real ◽  
Discontinuous Functions ◽  
Set Of Points

Let f(x) be a real function of the real variable x, let P be any point lying on the graph of f(x) and let l be a ray from P making an angle θ (− π < θ ≤ π) with the positive direction of the x-axis. We say that θ is a derivate direction of f(x) at the point P if the ray l meets the graph of f(x) in a set of points having a limit point at P.

scholarly journals Geometric and arithmetic postulation of the exponential function

1993 ◽  
Vol 54 (1) ◽  
pp. 111-127 ◽  
Author(s):  
J. Pila
Keyword(s):  
Exponential Function ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Real Variable ◽  
Mean Value ◽  
The Real ◽  
Mean Value Theorems ◽  
Auxiliary Functions

AbstractThis paper presents new proofs of some classical transcendence theorems. We use real variable methods, and hence obtain only the real variable versions of the theorems we consider: the Hermite-Lindemann theorem, the Gelfond-Schneider theorem, and the Six Exponentials theorem. We do not appeal to the Siegel lemma to build auxiliary functions. Instead, the proof employs certain natural determinants formed by evaluating n functions at n points (alternants), and two mean value theorems for alternants. The first, due to Pólya, gives sufficient conditions for an alternant to be non-vanishing. The second, due to H. A. Schwarz, provides an upper bound.

scholarly journals Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 113 ◽  
Author(s):  
Luciano Stefanini ◽  
Maria Letizia Guerra ◽  
Benedetta Amicizia
Keyword(s):  
Partial Order ◽  
Linear Space ◽  
Half Plane ◽  
Interval Analysis ◽  
General Setting ◽  
Real Variable ◽  
Single Variable ◽  
Variable Part ◽  
The Real ◽  
Interval Valued

We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact real intervals and we apply corresponding concepts for the analysis and calculus of interval-valued functions. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in calculus. Concepts related to convergence and limits, continuity, gH-differentiability and monotonicity of interval-valued functions are introduced and analyzed in detail. Graphical examples and pictures accompany the presentation. A companion Part II of the paper will present additional properties (max and min points, convexity and periodicity).

scholarly journals Further Notes on the Stieltjes Integral

1929 ◽  
Vol 1 (4) ◽  
pp. 234-240
Author(s):  
J. Hyslop
Keyword(s):  
Finite Interval ◽  
Real Variable ◽  
Stieltjes Integral ◽  
The Real ◽  
Previous Note ◽  
Real Functions ◽  
Finite Set ◽  
Image Position

The definition here used of the Stieltjes Integral is the same as that of a previous note, viz.:—Let f (x), φ (x) be two real functions defined in (a, b) a finite interval on the axis of the real variable x. Let Δ1, Δ2, …, Δn, be a finite set of sub-intervals which together make up (a, b). Δrφ denotes the increment of φ (x) in Δr. Let ξr be any point of Δr, and form the sum

Sign in / Sign up

Export Citation Format

Share Document