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

<pre>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}
]</pre>