@fedja I am very impressed by your proof, especially by a trick with logarithmic derivative. It looks OK. I am grateful to you.

@fedja Probably, there is a typo after (*) - one should have $T = 1 + (1-a)t$.

@fedja: About your phrase: "I'm also curious: what came first, the quadratic equation for x (or y) I wrote or your explicit formulas for x and z". Of course, the quadratic equation came first (z is discriminant) . Honestly speaking, both your proofs (by Iosif Pinelis and you) are looking rather unusual for me. But the idea to use the convexity (originally proposed by Iosif Pinelis and later used by you) looks as rather fruitful idea, while the appearance of T and F(T) in your proof looks as a rather unpredictable (and probably, not straightforward) step for me.

Also, it looks that better to use another letter, say u, instead of y, to avoid the confusing with y(t,a).

Thank you very much, but I need some time to verify all of that. The formula after the phrase "Plugging all that nonsense in, we see that our inequality is" needs adding " < 1". "

(Continuation.) It looks (from numerical/graphical analysis) that the function $y_{max}(a)$ ($0 < a < 1$) is monotonically increasing (in $a$) to $y_{max}(1-0) = 0.847...$, while the function $y_{\infty}(a)$ is monotonically decreasing to $y_{\infty}(1-0) = 0$.

The numerical analysis says that for fixed $a$ ($0 < a < 1$) the function $y(t,a)$ is monotonically increasing at interval $(0, t_0(a))$ to its maximal value $y(t_0(a),a) = y_{max}(a)$ and then it is monotonically decreasing at interval $(t_0(a), + \infty)$ to asymptotical value (which is a limit for $t \to + \infty$ ) $y_{\infty}(a) = \left(\frac{3 - 2a}{1 - a}\right)^{a - 3/2} 2^{2-a} < 1$.

Thank you very much! As far as I understand such trick (with algorithmical proof) can not be applied to initial function y(t,a). Moreover, I have suspicion that it is possible to prove finer estimate by this method, say y(t,a) < 0.9 (numerical calculations say about the bound near 0.848 ).

In physical literature the dual Coxeter number appears also in calculation of the so-called $\beta$-function for quantum Yang-Mills and QCD models corresponing to certain semi-simple (finite-dimensional) Lie groups. Sometimes it is denoted as $C_2$ (and related mistakenly to Casimir operator).

In case (b) we obtain: (c) $n_i = \sum_{j =1}^{r} B_{i j}$, $i = 1, \dots, r$, where $B = A^{-1}(I + P)$.

The sums $n_i = 2 \sum_{j =1}^{r} A^{-1}_{i j}$ ($i = 1,\dots, r$, where $r$ is rank of the Lie algebra) are integer numbers, which are components of twice dual Weyl vector in the basis of simple co-roots.

I agree with the latest comment of Prof. Humphreys (many thanks!). I have edited my original post.

Paul, thank you very much. (a) Indeed, $2A^{-1}$ for $D_{2n}$ is integer valued.