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

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