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

Abstract

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.

Simple efficient bounds for arcsine and arctangent functions

“…Inequalities in (5.5) are better than the corresponding inequalities in (5.4) in terms of sharpness. A more sharpened form of (5.4) and (5.5)can be seen in [3,5] which is given below. where β 1 = ln(2/π) ln(2/3) ≈ 1.113739.…”

On logarithm of circular and hyperbolic functions and bounds for exp(±x2)

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We show that certain known or new inequalities for the logarithm of circular hyperbolic functions imply bounds for exp(±x 2 ) proved in [1].

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New Refinements of Cusa-Huygens inequality

“…where α = ln(π/2)/ ln(3/2) ≈ 1.11374 and ζ = 1 are the best possible constants. Simple alternative proofs of (1.2) are offered in [3,4]. For other details about inequalities (1.1) and (1.2), we refer readers to [3-10, 13-18, 20-22].…”