A main inequality for several special functions

By using a recent generalization of the Cauchy–Schwarz inequality we obtain several new inequalities for some basic special functions such as beta and incomplete beta functions, gamma, polygamma and incomplete gamma functions, error functions, various kinds of Bessel functions, exponential integral functions, Gauss hypergeometric and confluent hypergeometric functions, elliptic integrals, moment generating functions and the Riemann zeta function. We also apply the generalized Cauchy–Schwarz inequality to some classical integral transforms. All these new inequalities obey the general form f2(x)≤k(x)f(px+q)f((2−p)x−q),f2(x)≤k(x)f(px+q)f((2−p)x−q), in which p,qp,q are real parameters and k(x)k(x) is a specific positive function.