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.
In this short review note we show that the new proof of Theorem 1.1 given by Zheng Jie Sun and Ling Zhu in the paper Simple proofs of the Cusa-Huygens-type and Becker-Stark-type inequalities is logically incorrect and present another simple proof of the same.Mathematics subject classification (2010): 26D05, 26D20.
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.
We rene Oppenheim's inequality as well as generalized Cusa-Huygens type inequalities established recently by some researchers. One of the results where the bounds of sin x / x are tractable will be used to obtain a sharp version of Yang's inequality.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.