Prove inequality of a convex function

In summary, to prove the inequality of a convex function, one typically demonstrates that for any two points \(x\) and \(y\) in its domain and any \(t\) in the interval [0, 1], the following holds: \(f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)\). This involves showing that the line segment connecting the points \((x, f(x))\) and \((y, f(y))\) lies above the graph of the function \(f\). The proof may leverage the definition of convexity, properties of derivatives, or Jensen's inequality, depending on the context and the specific function being analyzed.
  • #1
Lambda96
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Homework Statement
Proof that the following inequality holds
Relevant Equations
none
Hi,

I have problem to prove that the following inequality holds

Bildschirmfoto 2024-05-01 um 21.21.01.png

I thought of the following, since it is a convex function and applies, I started from the following inequality and transformed it further





Since the following applies it follows then follows i.e. the first part of the inequality

Is my approach correct, or does anyone have a better idea of how I can prove the inequality?
 
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  • #2
sounds like a condition about increasing functions, not convex functions. What is the definition of a convex function?

Also note your last step is wrong, you basically wrote down and and concluded
 
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  • #3
is convex if and only if

for all . You have arrived to the right conclusion, but it does not stem from monotonicity ( is convex but not monotone, for instance) nor the suspect step in between. Consider instead

  1. Why is this inequality true? (substitute )
  2. Check that it is equivalent to the first inequality.
 
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  • #4
Thank you Office_Shredder and nuuskur for your help 👍👍

Since it is a convex function and lies between and , I can represent as follows with

I then inserted this expression into the first part of the inequality

In order to get the required inequality, I applied the following, since it is a convex function
 
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