Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(−1/2, 1/2; 1; 1 − x c ) and F(−1/2 − δ, 1/2 + δ; 1;1 − x d ) on (0, 1) for given 0 < c ⩽ 5d/6 < ∞ and δ ∈ (−1/2, 1/2), and find the largest value δ 1 = δ 1(c, d) such that inequality F(−1/2, 1/2; 1; 1 − x c ) < F(−1/2 − δ, 1/2 + δ; 1; 1 − x d ) holds for all x ∈ (0, 1). Besides, we also consider the Gaussian hypergeometric functions F(a−1 −δ, 1-a+δ; 1;1 −x 3) and F(a−1, 1 −a; 1; 1−x 2) for given a ∈ [1/29, 1) and δ ∈ (a−1, a), and obtain the analogous results.