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Results tagged with inequalities
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user 40782
for questions involving inequalities, upper and lower bounds.
4
votes
3
answers
399
views
On some inequality (upper bound) on a function of two variables
There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables
$y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right …
- 371
1
vote
2
answers
228
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 …
- 371
0
votes
A real root of a cubic equation for a stationary point
In order to prove inequality (5) it is sufficient to prove two inequalities
$$V = 6(1-a)ap^2+(6-12a)p-6 > 0, \qquad (6V)$$
and
$$Z = 2(a-2)(a-1)ap^2+3(5a^2-9a+2)p+12a-18 > 0 \quad (6Z)$$
for all $x > … Due to Vieta's formulas and $ ap+1 >0$
we obtain the following inequalities:
$$3a(1-a)p^2+(3-2a)p+1 < 0, \qquad (18B)$$
$$4(a - 1)(a - 2)ap^3+ 6(-a^2- 3a+2)p^2 -24ap-12 > 0. …
- 371
0
votes
On polynomial equation of fourth order depending on two parameters and bound on a maximal root
The existence of the root $x_{*} > 1$ was proved above.
Now we prove the uniquenes of the root $x_{*} > 1$.
Let us suppose that there exists another root $x_{2,*} > 1$.
Without loss of generality …
- 371
0
votes
1
answer
133
views
On polynomial equation of fourth order depending on two parameters and bound on a maximal root
5a^2-9a+2)p +12a-18)x^2 \nonumber \\
- p^2 ((a-2)(4a^2 - 7a +1)p +9a^2-25a+18)x \nonumber \\
+ (a-2)(a-1)(2a-3)p^3 =0, \quad
(1)
\end{eqnarray}
where real parameters $a$ and $p$ obey inequalities … This task appears
in solving certain physical problem and here all inequalities have certain physical meanings. …
- 371