For the red trapezoid, you don't need to find the actual heights to find the average height. The average height is the height at the middle point, which is where the tangent line is referenced to.
@Jack_Callcott_AU
Жыл бұрын
@Keithfert490 Clever❕
@Eivindhamre
Жыл бұрын
I was going to comment the same. By finding the average height you need to simplify e^(½lnb+½lna)(lnb-lna) which is pretty simple
@atzuras
Жыл бұрын
Right, the convexity holds true for all R.
@PhaedkrK
Жыл бұрын
yep, typical Calc II concept
@leif1075
Жыл бұрын
WHY define the log mean thi s way --why notjust lna -ln b or lna + ln b or something else.
@allykid4720
Жыл бұрын
(sqrt(b) - sqrt(a))^2 >= 0 b + a - 2sqrt(ba) >= 0 (b+a)/2 >= sqrt(ab)
@pyrosanguineous
Жыл бұрын
5:32 If you notice that the limit is the reciprocal of the derivative definition of the natural logarithm at x=a, which is why the limit is 1/(1/a)
@plislegalineu3005
Жыл бұрын
That's why one could say it's almost like circular reasoning
@bigjazbo9217
Жыл бұрын
The log mean is ubiquitous in engineering, especially heat transfer and anything having to do with cylindrical shapes.
@insouciantFox
Жыл бұрын
I knew I'd seen it before
@Galileosays
Жыл бұрын
Indeed, I noticed the format too!
@davidruhdorfer3857
Жыл бұрын
I wanted to write exactly the same comment 👍
@robertmines5577
Жыл бұрын
Log Mean's actually have a lot of application to modeling transport phenomena when you have heat exchangers or separation membranes where two fluids can exchange heat or mass according to Newton's Law of Cooling or Fick's Law of Diffusion as they flow parallel to the surface separating the two. I don't remember the exact derivation, but it involves the fact that both of these processes show an exponential decay in concentration/temperature differences over time so the arithmetic average of temperature/concentration isn't as the logarithmic mean.
@JohnSmith-nx7zj
Жыл бұрын
Yep very familiar with the log mean as an engineer. Log mean temperature difference is used in sizing counter-current heat exchangers.
@ChefSalad
Жыл бұрын
You should do the arctangent mean. It's my favorite of the generalized means. It's defined as tan((1/n)*Σarctan(aₖ)), where the sum runs over all elements aₖ that you want to average. Its virtues include that it's well-defined for all real numbers, not just positives, and for all hyperreal numbers (which are basically the reals plus infinity), although infinitesimals are treated as having the infinitely small part being zero. This means that you can average infinities with actual numbers and get meaningful results. For example, the average of infinity and 0 is 1. The average of infinity, infinity, and 0 is √3.
@trueriver1950
11 ай бұрын
Nice! That's one I had not met before, and I like being able to take on infinity in a series when you are seeking a mean.
@Galileosays
Жыл бұрын
I wonder how the log-average for a series of points works out? The other two are (a1*a2*a3...*an)^(1/n) and (a1+a2+...an)/n
@Nolord_
Жыл бұрын
Exactly what I was wondering
@gp-ht7ug
Жыл бұрын
I like this kind of videos when geometric explanations are given in Cartesian axes
@lexinwonderland5741
Жыл бұрын
I wonder where it lies on the infinite scale of power means (harmonic
@falquicao8331
Жыл бұрын
Just by checking Desmos, it looks like it's always
@user-en5vj6vr2u
Жыл бұрын
There shouldn’t be one. Thatd be like saying x^a behaves as logx for some a
@__christopher__
Жыл бұрын
@@user-en5vj6vr2u Well, in some sense it does, for a=0.
@quazzydiscman
Жыл бұрын
"And we'll maybe derive this real quick just for completeness" This is how my wife wins arguments.
@bjornfeuerbacher5514
Жыл бұрын
Another expression which always gives a result which lies between the geometric and the arithmetic mean is (a + b + sqrt(ab))/3. It occurs e. g. when calculating the volume of a frustrum: That's given by its height times this "mean" of the area of the base and the area of the top. And I just found out that there is actually a name for this: It's called the "Heronian mean".
@minerscale
Жыл бұрын
Isn't that the arethmetic mean between the arethmetic mean and the geometric mean!?
@bjornfeuerbacher5514
Жыл бұрын
@@minerscale Actually, it's not a simple arithmetic mean of both, but a _weighted_ mean: 2/3 times the arithmetic mean plus 1/3 times the geometric mean.
@Nolord_
Жыл бұрын
Numerically, here's the inequalities between the means : arithmetic ≥ Heronian ≥ logarithmic ≥ geometric ≥ harmonic
@josea748
Жыл бұрын
@@Nolord_ so the Heronian is always above the log mean? Is the Heronian the smallest convex combination of am and gm above the log mean?
@Nolord_
Жыл бұрын
@@josea748 Maybe, that can depend on your definition of combination.
@broomstick0825
Жыл бұрын
Using mean value theorem gives us a more insight of how a value c within (a,b) is related with AM-GM inequality!
@bentationfunkiloglio
Жыл бұрын
One of my favorite videos. Not terrible complicated, but still really interesting.
@allenminch2253
Жыл бұрын
Such a cool video, thanks for sharing Michael! I loved the geometric elegance of the inequality proof!
@lock_ray
Жыл бұрын
Here is a generalization of this mean I thought of after watching this: L_a(x,y) = (a(x-y)/(x^a-y^a))^(1/(1-a)), taking appropriate limits when necessary. It reduces to the arithmetic mean when a=2, the logarithmic when a=0 and the geometric when a=-1. This raises the question, what about a=1? This would in a sense lie exactly between the arithmetic and logarithmic means. It turns out to become (x^x/y^y)^(1/(x-y)) / e, which I think could make for an interesting exercise.
@nodrogj1
Жыл бұрын
"Is there an in between mean?" Ah yes, the mean mean.
@xizar0rg
Жыл бұрын
*Central Limit Theorem has entered the chat*
@BikeArea
Жыл бұрын
Stop it! 😅
@__christopher__
Жыл бұрын
@@BikeArea That's mean! :-)
@aik21899
Жыл бұрын
You have heard of the elf on the shelf. Today we shall learn about the mean in-between.
@goodplacetostop2973
Жыл бұрын
14:59
@wolfmanjacksaid
Жыл бұрын
As an engineer I'm really glad to see a video about the log mean.
@gregsarnecki7581
Жыл бұрын
What about the AGM (arithmetic-geometric mean)? Doesn't it also lie between the arithmetic and geometric means? It's also related to the elliptic integrals and surely deserves more attention at the HS and undergraduate level.
@__christopher__
Жыл бұрын
It of course does lie between the arithmetic and geometric means by construction. Looking at it on Desmos, it seems to lie between the gemoetric mean and the logarithmic mean.
@H2Obsession
Жыл бұрын
Thanks Michael! I've been playing with various means you discussed (arithmetic, geometric, and harmonic) and also the RMS (root mean square). I always thought there must be an exponential or logarithmic mean and toyed with various ideas without success. Well you've answered that riddle with your video. I also like your proofs of the inequalities.
@phyarth8082
Жыл бұрын
AM-LM has practical application in physics in heat transfer physics problem.
@marc-andredesrosiers523
Жыл бұрын
Wish I had known about this proof much earlier in my life. 🙂
@GeoffryGifari
Жыл бұрын
Hmmm it is as if there is a set of two-argument functions called the "set of means", the "mean" function that belongs to it have the same properties as the arithmetic mean a+b/2, and each "mean" function of two real arguments a and b can be ordered
@benjamin7853
Жыл бұрын
Could this definition somehow be extended to more inputs? Like arithmetic mean is clearly (a+b+c)/3 and geometric mean is clearly cbrt(abc) but it doesn't seem obvious how this would work for the log mean
@JCCyC
Жыл бұрын
It can if the log mean is associative. Is it? Trying to solve that in my head but I'm getting logs of sums of logs ant it gets ugly.
@benjamin7853
Жыл бұрын
@@JCCyC well no because AM3(a, b, c) ≠ AM2(a, AM2(b, c)) if you know what i mean
@Tumbolisu
Жыл бұрын
With 4 inputs, both the arithmetic and geometric means of a, b, c and d are mean(mean(a, b), mean(c, d)). For instance, ((a+b)/2 + (c+d)/2)/2 = (a+b+c+d)/4 and √(√(ab)√(cd)) = 4thRoot(abcd). Let's try this with the logarithmic mean: Assume 0 < a < b < c < d. We have the two sub-means u := (b-a)/(ln(b)-ln(a)) and v := (d-c)/(ln(d)-ln(c)). Wait, was this the right thing to do? Maybe u should have been calculated from a and d, while v should be using b and c? Or maybe use a and c for u and b and d for v?? No matter what we do, we then need to figure out whether u or v is the larger one, and then calculate the mean of those two again. And now we have like, what, 6 possible cases to chew through? I feel like this doesn't generalize well...
@larsmarz3584
Жыл бұрын
An alternative way to show the initial inequality between the geometric and the arithmetic mean in by taking the logarithm and using Jensen's inequality.
@jackkalver4644
5 ай бұрын
In addition to an arithmetic mean and a geometric mean, any two numbers above 1 have what I call a double-geometric mean. If log_a (b)=log_b (c), b is the DGM of a and c. Means with other degrees of exponentiality are subjective to the base (although you could very well use e).
@yoav613
Жыл бұрын
Very nice!
@caryfitz
Жыл бұрын
Is it condition that the secant line is always "above" the tangent line a necessary and sufficient condition for convexity? Are there convex functions that don't have this property?
@EphemeralEphah
Жыл бұрын
Lovely video! I'd never heard of the logarithmic mean and deriving the inequality using areas was very interesting, your uploads are a bright point of my day. For completeness sake it might have been nice to first show that the log mean behaves as a mean (i.e. a
@MagicGonads
Жыл бұрын
I don't think means actually need that property (think for example of the average bearing, you can't order bearings in that way yet the mean has meaning), but yeah it would have been nice to show.
@EphemeralEphah
Жыл бұрын
@@MagicGonads You may be right, after (far too much) searching for a good definition, I couldn't actually find a well defined one for what "mean" actually refers to. Most dictionaries just define it as the arithmetic mean, while the closest real definitions I could find were vague at best, such as "a measure of central tendency" and Wikipedia's explanation that a "mean serves to summarize a group of data, often to better understand the overall value." I find it odd that it doesn't seem to have a well established mathematical definition.
@__christopher__
Жыл бұрын
Actually that's a consequence of the shown inequality: Since min(a,b)
@Risu0chan
Жыл бұрын
The log mean has other nice properties. It is homogeneous like other means: The mean of kx and ky is k times the mean of x and y. Thus you can scale the log mean to a function of one variable x/y. Also, it isn't difficult to show that L(x,y) < ((³√x+³√y)/2)³, the power mean of exponent 1/3. (which is less than the arithmetic mean).
@bjarnivalur6330
Жыл бұрын
I'm wondering, You can write the arithmetic mean like (t_1 + t_2 + ... + t_n) / (n) and the geometric mean as (t_1 t_2 ... t_n)^(1/n). Is there a similar way to put the logarithmic mean or does it only work if you have only two numbers?
@Francisco-vl5ub
Жыл бұрын
Have a look at generalised means (Wikipedia can get you started)
@bjarnivalur6330
Жыл бұрын
@@Francisco-vl5ub Thanks m8, will check it out
@popodori
Жыл бұрын
if am0 = arithmetic mean and gm0 = geometric mean, and am1 and gm1 the arithmetic and geometric mean of am0 and gm0, and repeat this a few times, then quickly am approaches gm and thus also the logarithmic mean, and this converges to something in the middle as well
@__christopher__
Жыл бұрын
While they approach each other, they don't approach the logarithmic mean. They approach the arithmetic-gemoetric eman, which is defined exactly by this limit.
@zathrasyes1287
Жыл бұрын
Great!
@Jack_Callcott_AU
Жыл бұрын
As usual, a very good video! FYI.. I have a very interesting little book from which I learned a lot called "Analytical Inequalities" by Nicholas D. Kazarinoff ; University of Michigan, HOLT RINEHART AND WINSTON 1961, but it doesn't seem to discuss the logarithmic inequality.
@Patapom3
Жыл бұрын
Amazing!
@Kapomafioso
Жыл бұрын
I don't think that complicated calculation of h1, h2 near the end was necessary. Because, (h1+h2)/2 (the arithmetic mean of the heights) is obviously just the middle point, so sqrt(ab). This is because they're connected with a straight line segment.
@ianfowler9340
Жыл бұрын
I've always been fascinated by the Harmonic Mean. Average Speed and Resistance in Parallel.
@stephenhamer8192
Жыл бұрын
Can we generalize the notion of a "mean" in another way? We note: i: x -> x is an increasing function, and the arithmetic mean, (a + b)/2 = inv i [ (i(a)+ i(b))/2 ] In is also an increasing function and we may write the geometric mean as sqrt (ab) = inv ln [ (in(a) + ln(b))/2 ] So let f be any increasing function (defined, say, on [0, inf), and define the "f-mean" of a, b in [0, inf), denoted M(f; a, b), to be inv f [ (f(a) + f(b)) / 2 ] Thus, M(x -> x^3; a, b) = cube-rt [ (a^3 + b^3) / 2 ] We note M(f; a, a) = a and for a < b, a < M(f; a, b) < b Does a "mean" have to have any other properties?
@stephenhamer8192
Жыл бұрын
And can the present mean be brought into this scheme? I.e., is there an increasing function, f, s.t.: f [ (b - a) /(ln b - In a) ] = [ f(a) + f(b) ] / 2 ?
@stephenhamer8192
Жыл бұрын
"Does a "mean" have to have any other properties?" Well, it has to be symmetric in its arguments
@stephenhamer8192
Жыл бұрын
Oh, and f, above, only has to be strictly monotonic: the harmonic mean uses f: x -> 1/x
@__christopher__
Жыл бұрын
Actually a lot of means are defined that way. For example, the harmonic mean is obtained with i(x) = 1/x. The root-mean-square is obtained with i(x) = x^2.And you can even get the geometric mean this way, using i(x) = ln(x).
@martinnyberg9295
Жыл бұрын
Is there a way to generalise the logarithmic mean that is useful and/or unambiguously “correct”, preserving both the inequality and that it should return “a” for the mean of arbitrarily many copies of “a”? 🤔
@lefty5705
Жыл бұрын
Since it's trapezoid, we can derive '(h1 + h2)/2 = e^(lnb+lna/2) = sqrt(ab)' direct
@danieleferretti9117
Жыл бұрын
How can be extended this logarithmic mean to a set of numbers? It seems not obvious to me
@ЕгорПетухов-ч8з
Жыл бұрын
You should rename the video "What does the logarithmic mean mean?"
@atzuras
Жыл бұрын
don't be mean
@Bromvolod
11 ай бұрын
Am I trying to understand this too soon? Because I'm struggling with it. As a chemistry "high schooler", I actually wonder if this is stuff that's usually taught at university or high school. I watched this because a teacher said "use the logarithmic mean for this because it's more precise" (related to heat exchange) and did not really explain why and what it is.
@Vadim_Ozheredov
Жыл бұрын
How to expand the definition to the arbitrary number of variables? I.e., a1, a2, a3 etc. instead a and b
@holyshit922
Жыл бұрын
5:32 Using L'Hopital ? why This is reciprocal of derivative of ln(x) at x=a so l'Hopital rule may produce circular reasoning
@fotnite_
Жыл бұрын
Because it's an indeterminate form. If you plug in x = a, you get 0/0, which is indeterminate.
@ib9rt
Жыл бұрын
But why did you take such a long and circuitous path to the pink area, rather than just writing it down on one or two lines from geometry? If (x, h) marks the midpoint of a straight line between points (ln(a), h1) and (ln(b), h2), then it follows from linearity that h = (h1 + h2)/2. Therefore the area of the trapezoid under the line is h(ln(b)-ln(a)). If h = exp[(ln(a) + ln(b))/2] (since h lies on the curve where the tangent is drawn), then immediately h = exp(ln(ab)/2) = sqrt(ab).
@BilalAhmed-wo6fe
Жыл бұрын
Very nice and elegent proof But can we use it in inequalities ?
@RussellSubedi
Жыл бұрын
But for two numbers a and b, we get GM(a, b) = GM(HM(a, b), AM(a, b)). We don't get the same for LM, i.e. LM(a, b) ≠ LM(GM(a, b), AM(a, b)). And that's kind of sad. I wonder what it'll converge to though.
@__christopher__
Жыл бұрын
If by "it" you mean taking arithmetic and geometric means repeatedly, the limit is, by definition, the arithmetic-geometric mean. This by construction has the property that AGM(AM(a,b), GM(a,b)) = AGM(a,b).
@rubensmith776
Жыл бұрын
12:24 Surely it's easier to integrate y between ln(b) and ln(a)?
@yb3604
Жыл бұрын
Awesome
@SuperKelam
Жыл бұрын
Isn't this the application of Hermite-Hadamard inequality for f = exp(x) from ln(a) to ln(b) ? THis is the simplest proof I've seen from this. Thanks!
@baronhannsz8900
Жыл бұрын
What about the Heronic Mean?
@bradhoward
Жыл бұрын
How do you do log mean of 3 or more numbers?
@JCCyC
Жыл бұрын
Yo Michael, benjamin7853 down there wondered if log mean can be extended to more than two operands. I realized it can if it's associative, but I tried to find that out in my head and it got gnarly. Good subject for a follow-up video?
@Fine_Mouche
Жыл бұрын
why you do the stuff only with 2 elements and not n elements ? cause it don't tell me if it's generalize to n-sqrt(sum(a_i)) =< sum(a_i)/n :/
@almightysapling
Жыл бұрын
These are some dank means
@Maths_3.1415
Жыл бұрын
Take a look at problem 5 of IMO 2022 And Problem 1 of IMO 2023 Both are number theory problems :)
@shruggzdastr8-facedclown
Жыл бұрын
So, is there a factorial mean?
@__christopher__
Жыл бұрын
Well, AFAIK the Gamma function (which generalizes the factorials) is strictly convex on the positive reals, which means the derivative is strictly increasing and thus invertible. Let's call the inverse function of that derivative I. Then according to the mean value theorem, the function I((Gamma(x)-Gamma(y))/(x-y)) is in between x and y. I guess one could call that the factorial mean of x and y,
@Neodynium.the_permanent_magnet
Жыл бұрын
That's interesting but I wonder if that log mean has any practical use.
@dneary
Жыл бұрын
For the area of the magenta area, you could also calculate the integral of the line: \sqrt{ab} \int_{\ln(a)}^{\ln(b)} x + 1 - \ln(\sqrt{ab}) dx = \sqrt{ab} [x^2/2 + x(1-\ln(\sqrt{ab})]_{\ln(a)}^{\ln(b)} = \sqrt{ab} ( \ln(b)^2/2 + \ln(b) - 1/2 \ln(b)(\ln(a) + \ln(b)) - \ln(a)^2/2 - \ln(a) + 1/2 \ln(a)(\ln(b) + \ln(a)) = \sqrt{ab}(\ln(b) - \ln(a)
@FrankHarwald
Жыл бұрын
Michael is just being mean today...
@zakiabg845
Жыл бұрын
At 1:36 you didn't subtract (a-b) ^2.
@casdinnissen6032
Жыл бұрын
You don't need to, because he's using an inequality and he's only makes the top of the fraction bigger, as (a-b)^2 is greater than 0
@zakiabg845
Жыл бұрын
@@casdinnissen6032 What's the factorial equality and unequality?
@casdinnissen6032
Жыл бұрын
@@zakiabg845 Im not sure what you mean with the factorial function. The inequality says that (ab)^(1/2)
@zakiabg845
Жыл бұрын
@@casdinnissen6032 I'm not talking about factorial fonction now.
@ianfowler9340
Жыл бұрын
In the pre-amble: WLOG, take a>=b [sqrt(a) - sqrt(b)]^2 >= 0. Expand and you are done. Short and sweet.
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