Absolutely (completely) monotonic functions and Jordan-type inequalities

Detailed analysis shows that a function f admits the double Jordan-type inequality if and only if f is analytic and even. Associated with f is the function g with f ( x ) = g ( x 2 ) . In this short note, based on this association, and using properties of absolutely and/or (completely) monotonic functions, we propose a concise method to derive the inequality from the coefficients in the Taylor’s series of f . The results include some existing ones as special cases.