Convexity properties related to Gauss hypergeometric function

We investigate the convexity property on $(0,1)$ of the functions φa,b,c and 1/φa,b,c, where φa,b,c(x)= c-log(1-x)/\,₂F₁(a,b,a+b,x), whenever a,b≥ 0 and a+b≤ 1. We Show that φa,b,c (respectively 1/φa,b,c) is strictly convex on $(0,1)$ if and only if c≤ -2γ-ψ(a)-ψ(b), (respectively c≥α₀) and φa,b,c (respectively 1/φa,b,c) is strictly concave on $(0,1)$ if and only if c≥ c(a,b) (respectively c∈[δ₋,δ₊]), where ψ 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,b∈[0,1], a+b≤ 1 and c≥ c(a,b) c+Γ(a)Γ(b)/Γ(a+b)≤ c-log(1-x)/\,₂F₁(a,b,a+b,x)+c-log(x)/\,₂F₁(a,b,a+b,1-x)≤(2c+2log 2)\,₂F₁(a,b;a+b;1/2), and for all x∈(0,1), a,b∈[0,1], a+b≤ 1 and c∈ [δ₋,δ₊] 1c+Γ(a+b)/Γ(a)Γ(b)≤ \,₂F₁(a,b,a+b,x)/c-log(1-x)+\,₂F₁(a,b,a+b,1-x)/c-log(x)≤\,₂F₁(a,b;a+b;1/2)(2c+2log 2).