Absolute monotonicity involving the zero-balanced hypergeometric function with applications

For a , b > 0 a,b>0 , let F ( x ) = F ( a , b ; a + b ; x ) F ( x ) =F ( a,b;a+b;x ) be the zero-balanced hypergeometric function. In this paper, we present the sufficient conditions for which the function x ↦ ± d d x x F ( x ) ln ⁡ ( 1 / ( 1 − x ) ) {equation*} x↦ ± d/dxxF ( x ) /ln ( 1/ ( 1-x ) ) {equation*} is absolutely monotonic on ( 0 , 1 ) ( 0,1 ) . As applications, we show the (absolute) monotonicity of the ratios F ( x 2 ) / F ( x ) F ( x² ) /F ( x ) and F ( 1 − x 2 ) / F ( 1 − x )