scholarly journals Some inequalities arising from analytic summability of functions

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3223-3230
Author(s):  
Sh. Saadat ◽  
M.H. Hooshmand
Keyword(s):  
Functional Equation ◽  
Holomorphic Function ◽  
Upper Bound ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Trigonometric Functions ◽  
Natural Numbers ◽  
Sums Of Powers ◽  
The Difference

Analytic summability of functions was introduced by the second author in 2016. He utilized Bernoulli numbers and polynomials for a holomorphic function to construct analytic summability. The analytic summand function f? (if exists) satisfies the difference functional equation f?(z) = f (z) + f?(z-1). Moreover, some upper bounds for f? and several inequalities between f and f? were presented by him. In this paper, by using Alzer?s improved upper bound for Bernoulli numbers, we improve those upper bounds and obtain some inequalities and new upper bounds. As some applications of the topic, we obtain several upper bounds for Bernoulli polynomials, sums of powers of natural numbers, (e.g., 1p+2p+3p+...+rp ? 2p! ?p+1 (e?r-1)) and several inequalities for exponential, hyperbolic and trigonometric functions.

Upper bounds for analytic summand functions and related inequalities

2021 ◽  
Vol 71 (5) ◽  
pp. 1103-1112
Author(s):  
Soodeh Mehboodi ◽  
M. H. Hooshmand
Keyword(s):  
Special Functions ◽  
Bernoulli Polynomials ◽  
Upper Bounds ◽  
Bernoulli Numbers ◽  
Natural Numbers ◽  
Bernoulli Polynomials And Numbers

Abstract The topic of analytic summability of functions was introduced and studied in 2016 by Hooshmand. He presented some inequalities and upper bounds for analytic summand functions by applying Bernoulli polynomials and numbers. In this work we apply upper bounds, represented by Hua-feng, for Bernoulli numbers to improve the inequalities and related results. Then, we observe that the inequalities are sharp and leave a conjecture about them. Also, as some applications, we use them for some special functions and obtain many particular inequalities. Moreover, we arrived at the inequality 1 p + 2 p + 3 p + ⋯ + r p ≤ 1 2 r p + 1 3 r p + 1 ( p + 1 ) + 2 3 p ! π p + 1 sinh ⁡ ( π r ) $1^p + 2^p + 3^p + \dots + r^p \leq \frac{1}{2}r^p + \frac{1}{3}\frac{r^{p+1}}{(p+1)} + \frac{2}{3}\frac{p!}{\pi^{p+1}}\sinh(\pi r)$ , for r sums of power of natural numbers, if p ∈ ℕ e and analogously for the odd case.

scholarly journals Obtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

2017 ◽  
Vol 9 (5) ◽  
pp. 73
Author(s):  
Do Tan Si
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Arithmetic Progression ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Arithmetic Progressions ◽  
Operator Calculus ◽  
Sums Of Powers ◽  
The Way

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials. 

scholarly journals Distribuion of Leading Digits of Numbers II

2019 ◽  
Vol 14 (1) ◽  
pp. 19-42
Author(s):  
Yukio Ohkubo ◽  
Oto Strauch
Keyword(s):  
Distribution Function ◽  
Real Number ◽  
Prime Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Natural Numbers ◽  
Increasing Functions

AbstractIn this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

Minimal complementation below uniform upper bounds for the arithmetical degrees

1996 ◽  
Vol 61 (4) ◽  
pp. 1158-1192
Author(s):  
Masahiro Kumabe
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Arithmetical Function ◽  
Natural Numbers

This paper was inspired by Lerman [15] in which he proved various properties of upper bounds for the arithmetical degrees. We discuss the complementation property of upper bounds for the arithmetical degrees. In Lerman [15], it is proved that uniform upper bounds for the arithmetical degrees are jumps of upper bounds for the arithmetical degrees. So any uniform upper bound for the arithmetical degrees is not a minimal upper bound for the arithmetical degrees. Given a uniform upper bound a for the arithmetical degrees, we prove a minimal complementation theorem for the upper bounds for the arithmetical degrees below a. Namely, given such a and b < a which is an upper bound for the arithmetical degrees, there is a minimal upper bound for the arithmetical degrees c such that b ∪ c = a. This answers a question in Lerman [15]. We prove this theorem by different methods depending on whether a has a function which is not dominated by any arithmetical function. We prove two propositions (see §1), of which the theorem is an immediate consequence.Our notation is almost standard. Let A ⊕ B = {2n∣n ∈ A} ∪ {2n + 1∣n + 1∣n ∈ B} for any sets A and B. Let ω be the set of nonnegative natural numbers.

scholarly journals Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

scholarly journals Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character χ

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 675 ◽  
Author(s):  
Serkan Araci ◽  
Waseem Khan ◽  
Kottakkaran Nisar
Keyword(s):  
Bernoulli Polynomials ◽  
Dirichlet Character ◽  
Bernoulli Numbers ◽  
Complex Order ◽  
Arbitrary Complex ◽  
Symmetric Identities

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed
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