Finding a solutions for an equation

precise evaluation – On some inequality (higher certain) on an obligation of two variables retort

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precise evaluation – On some inequality (higher certain) on an obligation of two variables

There is an issue (of somatic inception) which wants an analytical answer or a confidential. Let us assume the next actual-valued responsibility of two variables

$y (t,a) = 4 left(1 + frac{t}{x(t,a)}capable)^{ – a – 1/2}
left(1 – frac{2}{x(t,a)}capable)^{ 1/2} (x(t,a))^{-3/2} (z(t,a))^{1/4} left(1 + frac{t}{2}capable)^{ a},$

the place $t > 0$, $0 < a < 1$ and

$x(t,a) = frac{a-1}{2} t + frac{3}{2}+frac{1}{2}sqrt{z(t,a)}$,

$z(t,a) = (1-a)^2 t^2+ 2(3-a)t +9.$

It is needful to show that

$$y(t,a) < 1$$

for all $t > 0$ and $0 < a < 1$. The numerical evaluation helps this certain.

P.S. I apologize for too “technical” query. It appears that the inequality is precise too for $a = 1$
however it fails for $a > 1$.

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