On Jordan Type Inequalities for Hyperbolic Functions

has been in the focus of these studies and many refinements have been proved for it by Wu and Srivastava 9, 10 , Zhang et al. 11 , J.-L. Li and Y.-L. Li 5, 12 , Wu and Debnath 13–15 , Özban 16 , Qi et al. 17 , Zhu 18–29 , Sándor 30, 31 , Baricz and Wu 32, 33 , Neuman and Sándor 34 , Agarwal et al. 35 , Niu et al. 36 , Pan and Zhu 37 , and Qi and Guo 38 . For a long list of recent papers on this topic see 7 and for an extensive survey see 17 . The proofs are based on familiar methods of calculus. In particular, a method based on a l’Hospital type criterion for monotonicity of the quotient of two functions from Anderson et al. 39 is a key tool in these studies. Some other applications of this criterion are reviewed in 40, 41 . Pinelis has found several applications of this criterion in 42 and in several other papers. The inequality