# A minimizing problem involving Gauss hypergeometric functions

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?

• I don't understand the problem you are describing, but there is a good chapter on hypergeometric functions in M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Courier Dover Publications, New York, 1965. – user119324 Jan 28 '18 at 20:08
• @user119324 Yes, that is a nice book. But maybe I need a book which focuses more on the analysis of hypergeometric function as a real function. – Changwei Xiong Jan 28 '18 at 23:34