Recently I am considering a geometric question, which is reduced to the following problem.

Given $ L<0$ , let $ a\in [L/2,0]$ and $ b=L-a$ . For any $ c>0$ , let $ p,1-p$ solve $ $ x^2-x+c^2=0,$ $ and $ q,1-q$ solve $ $ x^2-x-c^2=0.$ $ Using Gauss hypergeometric functions, we define \begin{align*} f_a(c)&=(a^2-a)\left(\frac{F(1+p,2-p;2;a)}{F(p,1-p;1;a)}+\frac{F(1+q,2-q;2;a)}{F(q,1-q;1;a)}\right)\ &+(b^2-b)\left(\frac{F(1+p,2-p;2;b)}{F(p,1-p;1;b)}+\frac{F(1+q,2-q;2;b)}{F(q,1-q;1;b)}\right). \end{align*} Let $ c_0>0$ be the first positive zero of $ f_a(c)$ . So $ c_0$ depends on $ a$ , i.e. $ c_0=c_0(a)$ . To solve the geometric question I am considering, we need to prove $ $ \min_{a\in [L/2,0]} c_0(a)=c_0(0).$ $ I am a beginner to learn Gauss hypergeometric function. I suppose I need to understand those Gauss hypergeomertic functions appearing in $ f_a(c)$ , including their monotonicity, zeros, etc. Using Matlab I can get some rough pictures. But that is not enough for a rigorous proof. Could anyone recommend some text books concerning the problem above?