ganeshie8
  • ganeshie8
show that \[\large x^ty^{1-t}\le tx+(1-t)y\] \(x,y\gt 0\) and \(0\lt t\lt 1\)
Mathematics
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SOLVED
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chestercat
  • chestercat
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ParthKohli
  • ParthKohli
I can't help but notice the similarity between this and the parametric form of a complex number.
ganeshie8
  • ganeshie8
what parametric form.. polar ?
ParthKohli
  • ParthKohli
Nah. Never mind.

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ganeshie8
  • ganeshie8
I think you're referring to equation of straight line between two complex numbers in parametric form.. then there is some similarity yeah :)
anonymous
  • anonymous
Take logarithm ofboth sides, note that the log function is concave and rest is simply the statement of Jensen's inequality.
anonymous
  • anonymous
That is, \(t\log x + (1-t)\log y \leq \log (tx + (1-t)y)\)
ganeshie8
  • ganeshie8
Brilliant!
jtvatsim
  • jtvatsim
I think I found an "intuitive" proof using some basic Calculus... will post a pdf soon. :)
ganeshie8
  • ganeshie8
Please.. :) that geometric proof using Jensen's inequality is pretty neat, but im not really sure how popular that inequality is...
ganeshie8
  • ganeshie8
|dw:1439361518130:dw|
jtvatsim
  • jtvatsim
Oh wow! It's the same basic idea, except I did it for the exponential... LOL :)
jtvatsim
  • jtvatsim
See here.
1 Attachment
ganeshie8
  • ganeshie8
Interesting... please do share
ganeshie8
  • ganeshie8
That looks so neat! really a very neat substitution x/y = r !
jtvatsim
  • jtvatsim
It looks so contrived, but it arises naturally from playing with the inequality backwards. Assuming the inequality is true and trying to rearrange it leads to this idea. :)
ganeshie8
  • ganeshie8
Its not so contrived actually... we have \(x^ty^{1-t}\le tx+(1-t)y\) since \(y\gt 0\), dividing it through out gives \(x^ty^{-t}\le t\frac{x}{y}+(1-t)\) which is same as \(\left(\frac{x}{y}\right)^t\le t\left(\frac{x}{y}\right)+(1-t)\)
jtvatsim
  • jtvatsim
precisely! Which is exactly how I found it. :)
ganeshie8
  • ganeshie8
I see, thats all scratch work which goes in our heads... which we don't normally show in a rigorous proof..
anonymous
  • anonymous
This reminds me of gamma function
ikram002p
  • ikram002p
a generating function of gamma representation ive seen something like this

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