Absolutely (completely) monotonic functions and Jordan-type inequalities

Detailed analysis shows that a function ff admits the double Jordan-type inequality if and only if ff is analytic and even. Associated with ff is the function gg with f(x)=g(x2)f(x)=g(x2). 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 ff. The results include some existing ones as special cases.