Inequalities involving circular, hyperbolic and exponential functions

Bagul, Yogesh J.

doi:10.7153/jmi-2017-11-55

Cited by 8 publications

(5 citation statements)

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“…The inequalities (3.4), (3.5), and (3.6) have been obtained by putting p = 2 in Theorems 3.1, 3.2, and 3.3. These inequalities have already appeared in [2,3,10]. Next, we establish classical exponential bounds for the generalized hyperbolic functions.…”

Section: Resultsmentioning

confidence: 82%

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Some sharp circular and hyperbolic bounds of exp(-x2) with applications

Bagul¹,

Chesneau²

2020

APPL ANAL DISCRETE M

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This article is devoted to the determination of sharp lower and upper bounds for exp(−x 2 ) over the interval (− , ). The bounds are of the type a+f (x) a+1 α , where f (x) denotes either cosine or hyperbolic cosine. The results are then used to obtain and refine some known Cusa-Huygens type inequalities. In particular, a new simple proof of Cusa-Huygens type inequalities is presented as an application. For other interesting applications of the main results, sharp bounds of the truncated Gaussian sine integral and error functions are established. They can be useful in probability theory. * Corresponding author. Yogesh J. Bagul 2010 Mathematics Subject Classification. 26D07, 33B10, 33B20.

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Section: Resultsmentioning

confidence: 82%

“…In this regard, Chesneau [8,9] gave tight lower bounds of exp(x 2 ) over the real line. For some other sharp bounds, see [3,4], where the bounds are obtained over (0, 1) by the use of circular and hyperbolic functions. This type of bounds can in fact be obtained naturally over (0, π/2)(see [10]).…”

Section: Introductionmentioning

confidence: 99%

Some sharp circular and hyperbolic bounds of exp(-x2) with applications

Bagul¹,

Chesneau²

2020

APPL ANAL DISCRETE M

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“…The following two results concerning circular functions, they have been proved by C. Chesneau and Y. Bagul [1]. They improve theorems 1,2 of [2]. The first result concerns bounds for cos(x).…”

Section: A Variant Of the Bernoulli Inequalitymentioning

confidence: 97%

Sharp inequalities on circular and hyperbolic functions using a Bernoulli inequality type

Chouikha

2020

Preprint

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“…Recent developments can be found in [10,11,7,5,1,20,17,4,15,6,21,16,3,8,14,13,18,19] and the references therein. In this paper, we offer new simple tight (lower and upper) bounds involving these functions, with a high potential of interest for many researchers in mathematics or theoretical physics.…”

Section: Introductionmentioning

confidence: 99%

“…Note: To prove the inequalities (1.5), (1.6) and (1.7), we will simply use the results of [7,5,12]. We stress on the fact that it is not difficult to verify that all the results in [5] are also true in (0, π/2) with the respective best possible constants obtained accordingly (see [12]). Propositions 1.6 and 1.7 will be proved by the techniques of integration on some known results [4,6].…”

Section: Introductionmentioning

confidence: 99%

Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions

Bagul

Chesneau

2019

Cubo

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The prime goal of this paper is to establish sharp lower and upper bounds for useful functions such as the exponential functions, with a focus on exp(−x 2 ), the trigonometric functions (cosine and sine) and the hyperbolic functions (cosine and sine). The bounds obtained for hyperbolic cosine are very sharp. New proofs, refinements as well as new results are offered. Some graphical and numerical results illustrate the findings. RESUMENEl objetivo principal de este artículo es establecer cotas inferiores y superiores precisas para funcionesútiles tales como las funciones exponenciales, conénfasis especial en exp(−x 2 ), las funciones trigonométricas (coseno y seno) y las funciones hiperbólicas (coseno y seno). Las cotas obtenidas para el coseno hiperbólico son muy precisas. Se presentan, tanto nuevas demostraciones y refinamientos, como resultados nuevos.Algunos resultados numéricos y gráficos ilustran los resultados encontrados.

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Inequalities involving circular, hyperbolic and exponential functions

Abstract: Abstract. This paper is aimed at obtaining some new lower and upper bounds for the functions cos x , sinx/x , x/ cosh x , thus establishing inequalities involving circulr, hyperbolic and exponential functions.Mathematics subject classification (2010): 26D05, 26D07.

Cited by 8 publications

References 1 publication

Some sharp circular and hyperbolic bounds of exp(-x2) with applications

Some sharp circular and hyperbolic bounds of exp(-x2) with applications

Sharp inequalities on circular and hyperbolic functions using a Bernoulli inequality type

Some New Simple Inequalities Involving Exponential, Trigonometric and Hyperbolic Functions

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