A generic inequality for basic special functions

We introduce a generic inequality of the form f(2)(x) <= f(s)f (2x-s) and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions arcsin x and arctan x.