For $f : I mapsto mathbb{R}$ convex and of class $C^2$, if $f' geq m$, show an integral inequality regarding...
$begingroup$
Let be $I$ an interval of $mathbb{R}$, $f : I to mathbb{R}$ a convex function of class $C^2$ such that $forall t in I, f'(t) geq m$.
I want to show that:
$begin{equation*}
forall (a, b) in I^2, left lvert displaystyle int_a^b exp left[i f(t)right] textrm{d}t right rvert leq dfrac{2}{m}
end{equation*}$
So my approach was: we have an idea of the speed of $f$ due to its lower bound and its convexity so then we can get, using fundamental theorem of analysis and Taylor theorems get bounds on how much does it spin in the unit circle and extrapolates "its area" (does it makes sense here?).
But I am unable to get anything near the result, I guess, the simplest thing is to work on $I = int_a^b cos (f(t)) textrm{d}t$ beforehand, but I have no idea how to tackle this.
I'd welcome very much any hint rather than a complete solution in order to get rather real understanding of those kind of problems.
real-analysis integration convex-analysis
asked Dec 17 '18 at 21:55
RaitoRaito
686415
$endgroup$
add a comment |
$begingroup$
Let be $I$ an interval of $mathbb{R}$, $f : I to mathbb{R}$ a convex function of class $C^2$ such that $forall t in I, f'(t) geq m$.
I want to show that:
$begin{equation*}
forall (a, b) in I^2, left lvert displaystyle int_a^b exp left[i f(t)right] textrm{d}t right rvert leq dfrac{2}{m}
end{equation*}$
So my approach was: we have an idea of the speed of $f$ due to its lower bound and its convexity so then we can get, using fundamental theorem of analysis and Taylor theorems get bounds on how much does it spin in the unit circle and extrapolates "its area" (does it makes sense here?).
But I am unable to get anything near the result, I guess, the simplest thing is to work on $I = int_a^b cos (f(t)) textrm{d}t$ beforehand, but I have no idea how to tackle this.
I'd welcome very much any hint rather than a complete solution in order to get rather real understanding of those kind of problems.
real-analysis integration convex-analysis
asked Dec 17 '18 at 21:55
RaitoRaito
686415
$endgroup$
$begingroup$
For that simple result, try to use the 2nd integral mean value theorem, since $f'$ is monotonic due to the convexity of $f$.
$endgroup$
– xbh
Dec 18 '18 at 0:31
add a comment |
$begingroup$
Let be $I$ an interval of $mathbb{R}$, $f : I to mathbb{R}$ a convex function of class $C^2$ such that $forall t in I, f'(t) geq m$.
I want to show that:
$begin{equation*}
forall (a, b) in I^2, left lvert displaystyle int_a^b exp left[i f(t)right] textrm{d}t right rvert leq dfrac{2}{m}
end{equation*}$
So my approach was: we have an idea of the speed of $f$ due to its lower bound and its convexity so then we can get, using fundamental theorem of analysis and Taylor theorems get bounds on how much does it spin in the unit circle and extrapolates "its area" (does it makes sense here?).
But I am unable to get anything near the result, I guess, the simplest thing is to work on $I = int_a^b cos (f(t)) textrm{d}t$ beforehand, but I have no idea how to tackle this.
I'd welcome very much any hint rather than a complete solution in order to get rather real understanding of those kind of problems.
real-analysis integration convex-analysis
asked Dec 17 '18 at 21:55
RaitoRaito
686415
$endgroup$
Let be $I$ an interval of $mathbb{R}$, $f : I to mathbb{R}$ a convex function of class $C^2$ such that $forall t in I, f'(t) geq m$.
I want to show that:
$begin{equation*}
forall (a, b) in I^2, left lvert displaystyle int_a^b exp left[i f(t)right] textrm{d}t right rvert leq dfrac{2}{m}
end{equation*}$
So my approach was: we have an idea of the speed of $f$ due to its lower bound and its convexity so then we can get, using fundamental theorem of analysis and Taylor theorems get bounds on how much does it spin in the unit circle and extrapolates "its area" (does it makes sense here?).
But I am unable to get anything near the result, I guess, the simplest thing is to work on $I = int_a^b cos (f(t)) textrm{d}t$ beforehand, but I have no idea how to tackle this.
I'd welcome very much any hint rather than a complete solution in order to get rather real understanding of those kind of problems.
real-analysis integration convex-analysis
real-analysis integration convex-analysis
asked Dec 17 '18 at 21:55
RaitoRaito
686415
asked Dec 17 '18 at 21:55
RaitoRaito
686415
asked Dec 17 '18 at 21:55
RaitoRaito
686415
asked Dec 17 '18 at 21:55
RaitoRaito
686415
asked Dec 17 '18 at 21:55
RaitoRaito
686415
686415
$begingroup$
For that simple result, try to use the 2nd integral mean value theorem, since $f'$ is monotonic due to the convexity of $f$.
$endgroup$
– xbh
Dec 18 '18 at 0:31
add a comment |
$begingroup$
For that simple result, try to use the 2nd integral mean value theorem, since $f'$ is monotonic due to the convexity of $f$.
$endgroup$
– xbh
Dec 18 '18 at 0:31
$begingroup$
For that simple result, try to use the 2nd integral mean value theorem, since $f'$ is monotonic due to the convexity of $f$.
$endgroup$
– xbh
Dec 18 '18 at 0:31
$begingroup$
For that simple result, try to use the 2nd integral mean value theorem, since $f'$ is monotonic due to the convexity of $f$.
$endgroup$
– xbh
Dec 18 '18 at 0:31
add a comment |
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$begingroup$
For that simple result, try to use the 2nd integral mean value theorem, since $f'$ is monotonic due to the convexity of $f$.
$endgroup$
– xbh
Dec 18 '18 at 0:31