Convexity properties related to Gauss hypergeometric function

We investigate the convexity property on (0, 1) of the functions ₀, ₁, ₂ and 1/₀, ₁, ₂, where ₀, ₁, ₂ (x) = c- (1-x) \, ₂F₁ (a, b, a+b, x), whenever a, b 0 and a+b 1. We Show that ₀, ₁, ₂ (respectively 1/₀, ₁, ₂) is strictly convex on (0, 1) if and only if c -2- (a) - (b), (respectively c₀) and ₀, ₁, ₂ (respectively 1/₀, ₁, ₂) is strictly concave on (0, 1) if and only if c c (a, b) (respectively c_-, _+), where is the Polygamma function. This generalizes some problems posed by Yang and Tian and complete the study of convexity properties of functions studied by the author in bouali. As applications of the convexity and concavity, we establish among other inequalities, that for all x (0, 1), a, b0, 1, a+b 1 and c c (a, b) c+ (a) (b) (a+b) c- (1-x) \, ₂F₁ (a, b, a+b, x) +c- (x) \, ₂F₁ (a, b, a+b, 1-x) (2c+2 2) \, ₂{F₁ (a, b;a+b;1/2) }, and for all x (0, 1), a, b0, 1, a+b 1 and c _-, _+ 1c+ (a+b) (a) (b) \, ₂F₁ (a, b, a+b, x) c- (1-x) +\, ₂F₁ (a, b, a+b, 1-x) c- (x) \, ₂F₁ (a, b;a+b;1/2) (2c+2 2).