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 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. Tags equation, functional ### content

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