Jensen's Inequality: How to Use It

This video is dedicated to introducing Jensen's inequality and applications of it to establishing inequalities of various kinds. We use it to prove the arithmetic geometric mean inequality and also establish another olympiad-like inequality with it.

[@kenthedawg6383]

Using Jensen for AM-GM was slick! Great video Prof Omar. Your explanations were very clear and easy to follow. I also like this new format.

[@kelvinsenteza]

Thanks a lot professor - I've been trying to wrap my head around Jensen's inequality, and the video was exactly what I needed!

[@brendanchamberlain9388]

Super cool. I remember one of your other videos on an inequality used Jensen's Inequality to solve it. You did a good job of explaining it then, but its really nice to have a dedicated video for it, thanks for sharing.

[@ProfOmarMath]

Thanks Brendan!

[@MyOneFiftiethOfADollar]

Drawing the concave down graph with midpoints really helped to see where the constants in Jensen's inequality originate. The square root function is also concave, negative second derivative, and can be used to prove the well known two variable special case

[@ProfOmarMath]

Definitely!

[@missionjoshi1362]

The second derivative of f we get is -2S/(x+s)^3 from this can we say that f''(x)

[@erickarwa-0705]

You do an amazing work. Thank you.

[@orenawaerenyeager]

Perfectly explained
Thanks 👍

[@yaseengarehmohammadlou9349]

Thanks professor omar for this beautiful lesson which is more beautiful with your great explanation

[@ProfOmarMath]

Thanks Yaseen!

[@technolan6015]

we may solve that inequality by multiplying both sides by (1+ab)(1+ac)(1+bc), and with a little bit of factorizing and simplifying, we get that we need to prove that 2(a^2 + b^2 + c^2)>= ab + ac + bc, which is possible

[@yousuf_w1]

Thanks a lot professor
I’ve been watching your video for a year ( when I started preparing for the IMO)
I am now a 7th grader
Hopefully I will be able to be a medalist this year
Last year I was hardly chosen to be in the team because of I was 6th grader and I got a little bit higher than the other (less than the other team members)

[@ProfOmarMath]

That’s great Yousuf!

[@samegawa_sharkskin]

thank you so much!! the text explanations were so confusing, your explanations makes much more sense :)

[@ProfOmarMath]

Anytime!

[@isaiaferrario]

Thank you for this video! Very interesting!

[@ProfOmarMath]

Thanks Isaia!

[@jantavihar07bhalswa52]

Thanks professor for such a beautiful content and motivating other people to feel mathematics love from India will meet you one day!!!👍😊☺️👍

[@ProfOmarMath]

Thanks Janta!

[@amberdeshbhratar9342]

Amazing proof of AM - GM inequality.

[@ProfOmarMath]

Isn't it cool Amber!

[@user-wi1rj4iw9y]

Thank you! 谢谢！

[@pokoknyaakuimut001]

I love this channel 🥺🙃

[@ProfOmarMath]

Thank you for being a part of it!

[@joelgerlach9406]

In the last example how can we fix s? It is still dependent on a or b or c, so it changes as x changes

[@Sgh591]

Thanks a lot professor ! 😀

[@ProfOmarMath]

Thank you!

[@prithujsarkar2010]

Hi professor, can you make a video on weighted inequalities such as weighted jensen, am-gm, etc etc

[@ProfOmarMath]

Neat idea!

[@yashvardhan2093]

Seeing the 1s in the numerator I used Cauchy Schwarz and it worked
Nice video

[@alexwang982]

How

[@geometrydashmega238]

Cool inequalities

[@coldsoup49]

Cool example!

[@ProfOmarMath]

Isn't it? 😍

[@djvalentedochp]

nice video I have a lot to learn

[@ProfOmarMath]

Feel free to ask anything, I'm happy to be a part of your learning process

[@CarlosMarante]

Very nice video. Made me eager to look for the others, nevertheless the layout you used in most of the other videos makes the readability very hard. It would be better to use the full screen for the writing, and if you want to appear in the video, to do so on a smaller inset in a corner. Best.

[@ProfOmarMath]

Thanks for this suggestion, I actually really appreciate it!

[@user-iu9rg8ek4z]

nice! from JPN,

[@ualibek5666]

I wonder what the equal conditions are. The question is will the inequality ever be an equality? If so, when?

[@Ayush-mg6xw]

Sir but i didn't understand why a+b+c=s is fixed value is it give!n?

[@SupposeMath]

How f(a) =a/(a+abc)?

[@ProfOmarMath]

The idea is we hold a+b+c constant and then create a function and analyze it. Since a+b+c=abc that means abc is constant too

[@fedryfirman.a5783]

i'm sorry but i don't get it why s/3 is equal to 1/4, thanks

[@samsunggallaxy1647]

😘😘

[@AkashSharma-kr5dy]

Something is wrong in this question, the inequality should be of opposite direction.

[@AkashSharma-kr5dy]

You can check the inequality by putting ac=ab=BC=0.

[@chamsderreche5750]

positive=the reals strictly greater than 0
nonnegative=the reals greater than or equal to 0
this confused me for a VERY long time

[@hclaptop4633]

cannot see the last part f***