Suppose $F$ is the Gaussian hypergeometric function 2F1, $x\in \mathbb{N}$, $t>x \in \mathbb{N}$, $\mu \in (0,1)$. Is the function: $$f(\mu)=\mu^{x+1} F(x,-t+x,2+x,\mu),$$ convex in $\mu$? I've done some numerical examples that seem to verify this with the following manipulate Mathematica code (apologies for the mess):

Manipulate[ Plot[ { Subscript[\[Mu], 1]^(x + 1) b* Hypergeometric2F1[x, -t + x, 2 + x, Subscript[\[Mu], 1]] + (1 - b) Subscript[\[Mu], 2]^(x + 1)* Hypergeometric2F1[x, -t + x, 2 + x, Subscript[\[Mu], 2]], (b*Subscript[\[Mu], 1] + (1 - b) Subscript[\[Mu], 2])^(x + 1) Hypergeometric2F1[ x, -t + x, 2 + x, (b*Subscript[\[Mu], 1] + (1 - b) Subscript[\[Mu], 2]) ] }, {b, 0, 1} ], {t, 1, 80, 1}, {x, 1, 80, 1}, {Subscript[\[Mu], 1], 0, 1}, {Subscript[\[Mu], 2], 0, 1} ]

EDIT: Forgot to add $x+1$ instead of $x$.