Inequalities involving circular, hyperbolic and exponential functions
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Inequalities involving circular, hyperbolic and exponential functions
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J ournal of Mathematical I nequalities Volume 11, Number 3 (2017), 695–699
doi:10.7153/jmi-2017-11-55
INEQUALITIES INVOLVING CIRCULAR, HYPERBOLIC AND EXPONENTIAL FUNCTIONS
YOGESH J. BAGUL
Dedicated to my mothers Vimal and Vandana
(Communicated by E. Neuman)
Abstract. This paper is aimed at obtaining some new lower and upper bounds for the functions cos x , sin x/x , x/ cosh x , thus establishing inequalities involving circulr, hyperbolic and exponential functions.
1. Introduction The well-known Jordan’s inequality [1], [6] is stated as follows: 2 sin x < 1, < π x
x ∈ (0, 1).
(1)
During the past few years the Jordan’s inequality has been in the focus of studies on the trigonometric and hyperbolic inequalities and many refinements have been proved [1–3], [5–11]. R. Kl´en, M. Visuri, and M. Vuorinen [7, Thm. 1.2] have proved the inequality
