Shafer–Fink type inequalities for arc lemniscate functions

Abstract

In this paper, we investigate the monotonicity and inequalities for some functions involving the arc lemniscate and the hyperbolic arc lemniscate functions. In particular, sharp Shafer-Fink type inequalities for the arc lemniscate and the hyperbolic arc lemniscate functions are proved.Keywords. arc lemniscate functions, hyperbolic arc lemniscate functions, lemniscate functions, hyperbolic lemniscate functions, Shafer-Fink type inequalities Mathematics Subject Classification (2010). 26D07, 33E05

Some general Wilker-Huygens inequalities

“…The monotonicity of f /x and g/x can be easily proved by Lemma 6 for (f, g) ∈ S 1 and had been treated in [30,Lemma 3.1] for (f, g) ∈ S 2 . In order to present more clearly, we summarize all the monotonicity of f /x and g/x in Tabel 1 while the monotonicity of L f,g (x) can be found in Propositions 10-17.…”