Sharp Cusa type inequalities for trigonometric functions with two parameters

2014/08/10 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #26D15 #33B10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 26D05 #Secondary 26A48 #math.CA #msc:26A48 #msc:26D05 #msc:26D15 #msc:33B10

paper · pdf · doi:10.48550/arxiv.1408.2250

29 pages

arxiv created 2014/08/10 · arxiv updated 2014/08/12

Abstract

Let ( p,q) ↦ β( p,q) be a function defined on ℝ². We determine the best or better p,q such that the inequality% ( (sin x)/(x)) ^p<( >) 1-β( p,q) +β( p,q) cos ^qx% holds for x∈ ( 0,π/2) , and obtain a lot of new and sharp Cusa type inequalities for trigonometric functions. As applications, some new Shafer-Fink type and Carlson type inequalities for arc sine and arc cosine functions, and new inequalities for trigonometric means are established.

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