Some sharp Wilker type inequalities and their applications

2013/04/19 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #26D05 #26D15 (Secondary) #33B10 (Primary) 26A48 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:26A48 #msc:26D05 #msc:26D15 #msc:33B10

paper · pdf · doi:10.48550/arxiv.1304.5392

15 pages

arxiv created 2013/04/19 · arxiv updated 2013/04/22

Abstract

In this paper, we prove that for fixed k≥ 1, the Wilker type inequality equation* (2)/(k+2)((sin x)/(x)) ^kp+(k)/(k+2)((% tan x)/(x))^p>1 equation*% holds for x∈ (0,π/2) if and only if p>0 or p≤ -% (ln (k+2) -ln 2)/(k(ln π-ln 2)). It is reversed if and only if -(12)/(5(k+2))≤ p<0. Its hyperbolic version holds for x∈ (0,∞) if and only if % p>0 or p≤ -(12)/(5(k+2)). And, for fixed k<-2, the hyperbolic version is reversed if and only if p<0 or p≥ -(12)/(% 5(k+2)). Our results unify and generalize some known ones.

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