Inequalities involving hypergeometric and related functions

An inequality is being proved which is connected to cost-effective numerical density estimation of the hyper-gamma probability distribution. The left-hand side of the inequality is a combination of two in the third parameter distinct versions of the hypergeometric function at the point one. All three parameters are functions of the distribution’s terminal shape. The first and second are equal. The distinct third parameters of the two hypergeometric functions depend on terminal and initial shape. The other side of the inequality is determined by the quotient of two infinite series, which are related to the first derivatives with respect to terminal shape of the hypergeometric functions which appear in its left-hand side.