An affirmative answer to an open problem on Ramanujan's asymptotic formula of zero-balanced hypergeometric function

In this paper, by using recurrence method and new properties of beta function and Ramanujan $R$-function, we prove the absolute monotonicity result for a function related to Ramanujan asymptotic formula of zero-balanced hypergeometric function, which gives an affirmative answer to an open problem proposed by Anderson Vamanamurthy and Vuorinen \cite{Anderson-CIIQM-JW-1997} in 1997.