Inequalities for generalized trigonometric and hyperbolic sine functions

2012/12/05 by Miao-Kun Wang, Wang, Miao-Kun, Yu-Ming Chu +3
Mathematics · #33B10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #G.1.2 #acm:33B10 #math.CA #msc:33B10

paper · pdf · doi:10.48550/arxiv.1212.4681

7 pages

arxiv created 2012/12/05 · arxiv updated 2012/12/20

Abstract

We prove that the inequalities sin_p,q(√(rs))≥ √sin_p,q(r)sin_p,q(s) and \sinh_p,q(√(r^*s^*)) ≤ √\sinh_p,q(r^*)\sinh_p,q(s^*) hold for all p,q∈(1,∞), r,s∈(0,∫₀¹(1-t^q)^-1/pdt) and r^*,s^*∈(0,∫₀^∞(1+t^q)^-1/pdt), where sin_p,q and \sinh_p,q are the generalized trigonometric and hyperbolic sine functions, respectively. As a consequence of the results, we prove a conjecture due to Bhayo and Vuorinen [J. Approx. Theory, 164(2012)].

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