Refinements of Mitrinovi\'c-Cusa inequality

Abstract

The Mitrinovi\'c-Cusa inequality states that for x\in(0,{\pi}/2) (cos x)^{1/3}<((sin x)/x)<((2+cos x)/3) hold. In this paper, we prove that (cos x)^{1/3}<(cos px)^{1/(3p^{2})}<((sin x)/x)<(cos qx)^{1/(3q^{2})}<((2+cos x)/3) hold for x\in(0,{\pi}/2) if and only if p\in[p_{1},1) and q\in(0,1/\surd5], where p_{1}=0.45346830977067.... And the function p\mapsto(cos px)^{1/(3p^{2})} is decreasing on (0,1]. Our results greatly refine the Mitrinovi\'c-Cusa inequality.