Let \(\lambda >0\) and \(\mathcal {K}_a\) denote the generalized elliptic integral of the first kind. Previous studies have confirmed the monotonicity and convexity of the function \(x\mapsto \mathcal {K}_a(\sqrt{x})/\log (1+\lambda /\sqrt{1-x})\) on the interval (0, 1), where \(\log (1+\lambda /\sqrt{1-x})\) serves as a non-Ramanujan asymptotic function–distinct from the traditional Ramanujan asymptotic forms typically associated with such special functions–for the zero-balanced hypergeometric function. In this paper, these properties are extended to the zero-balanced hypergeometric function. Specifically, we establish the monotonicity and convexity of the function of , as well as the concavity of its reciprocal, all defined on (0, 1). This result also provides a positive verification of a conjecture proposed by Yang and Tian when \(a=b=1/2\) .
