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In the note, some new proofs for inequalities involving trigonometric functions aregiven. 1. Introduction In [10], J. B. Wilker proposedthat(a) If 0 2 . (1)(b) There exists a largest constant c such that sin xx 2 +tan xx> 2+ cx 3 tan x (2)for 0 sin xx 2 +tan xx> 2+2π 4 x 3 tan x. (3)Theconstants 845 and 2π 4 arebestpossible,thatis,theycannotbereplacedbysmalleror larger numbersrespectively.The inequalitiesin (1) and (3) are called Wilker’s inequalitiesin [4].In this note, we will give new proofs for the inequalities in (1) and (3). Mathematics subject classification (2000): Primary 26D05. Key words and phrases : Trigonometric function, Wilker’s inequality, Bernoulli number. The first and third authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth ofHenan Province (#0112000200), SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800),SF for Pure Research of Natural Science of the Education Department of Henan Province(#1999110004), Doctor Fund ofJiaozuo Institute of Technology, China.. c |
