February 29th, 2012, 09:53 AM  #1 
Newbie Joined: Nov 2009 Posts: 16 Thanks: 0  Functional equation
Here is my problem: There is given the functional equation: for . We also know that and . Additionally, we assume the continuity and strict monotonicity of . Is it possible to get any information on ? In particular, can we prove that ? Thank you for hints and help. 
February 29th, 2012, 10:38 AM  #2 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Functional equation
Show that the only function satisfying that functional equation and the given information about the function is , thus proving that .

February 29th, 2012, 11:50 AM  #3 
Newbie Joined: Nov 2009 Posts: 16 Thanks: 0  Re: Functional equation
OK. However, I would be grateful for any tips how to prove it. The equation which I wrote is a derivative of the Hosszu functional equation (solution of this equation under some assumption is a linear function). But Hosszu equation involves two variables, not one. How to handle with this?

February 29th, 2012, 12:04 PM  #4 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Functional equation
I am not familiar with the Hosszu functional equation. What is the equation, and under what assumption is the equation a linear function? Because essentially what you have to prove is that the functional equation you gave () must represent a linear function.

February 29th, 2012, 12:57 PM  #5 
Newbie Joined: Nov 2009 Posts: 16 Thanks: 0  Re: Functional equation
The Hosszu functional eq. is . If is continuous and satisfies this eq., then f is linear. Do you maybe have any idea how to solve an original problem? Best regards. 
February 29th, 2012, 01:20 PM  #6 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Functional equation
One more question about the Hosszu functional equation: does the function have to satisfy that equation for all x and all y in order to be linear?

February 29th, 2012, 11:37 PM  #7 
Newbie Joined: Nov 2009 Posts: 16 Thanks: 0  Re: Functional equation
If I knew this, I wouldn't ask. So far, I haven't found any results on Hosszu equation on restricted domain, so in my opinion the equation must be satisfied for all x and y and then the function is linear.

March 1st, 2012, 10:01 AM  #8 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Functional equation
You can use the functional equation recursively as follows: Set x = 1/4. Then you get , or (1) . Then set x = 1/8 to yield + f(5/" /> and set x = 3/8 to yield + f(7/" />. Adding these two equations yields + f(3/ + f(5/ + f(7/" />, and substituting from (1) yields + f(3/ + f(5/ + f(7/" />. By using this process recursively, you can show that and thus = . Now, using the functional equation again, we have and thus . Hence . I'm not sure if you can do anything with that, but that's my attempt at doing something useful with this problem. 

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