One answer to an open problem on the monotonicity of Gaussian hypergeometric functions with respect to parameters

The authors prove that for arbitrarily given b∈(0,∞), r∈(0,1)and n∈N0, and for λ∈R, the functions a↦F(a,b;a+b;r)aλanda↦F(a,b;a+b;r)-∑k=0n(a,k)(b,k)(a+b,k)k!rkaλare both strictly increasing (decreasing) on (0,∞)if and only if λ≤0(λ≥max{1,b}, respectively), where F(a,b;a+b;r)denotes the Gaussian hypergeometric function. This result gives one answer to an open problem raised by Qiu et al. (J Math Anal Appl 2020, 492(2):31, https://doi.org/10.1016/j.jmaa.2020.124469) recently.