For given \(a,b,c,d\in (0,\infty)\), \(a\leq b\), and \(c< d\), the authors study the problem of comparing the Gaussian hypergeometric functions \(F(a,b; a+b; 1-x^c)\) and \(F(a-\delta, b+\delta; a+b; 1- x^d)\). In particular they solve some open problems by determining \(\sup\{\delta\in (0,\infty): F(a,b; a+b; 1-x^c)< F(a-\delta, b+\delta; a+b; 1-x^d)\), \(x\in (0,1)\}\).