My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
February 29th, 2012, 10:53 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.
Nobody1111 is offline  
 
February 29th, 2012, 11:38 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 .
icemanfan is offline  
February 29th, 2012, 12:50 PM   #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?
Nobody1111 is offline  
February 29th, 2012, 01:04 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.
icemanfan is offline  
February 29th, 2012, 01:57 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.
Nobody1111 is offline  
February 29th, 2012, 02:20 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?
icemanfan is offline  
March 1st, 2012, 12:37 AM   #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.
Nobody1111 is offline  
March 1st, 2012, 11:01 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.
icemanfan is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
equation, functional



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
A Functional Equation Pell's fish Real Analysis 2 February 3rd, 2014 04:08 AM
functional equation nukem4111 Real Analysis 1 September 16th, 2013 04:44 AM
Functional equation frobenius Math Events 4 June 6th, 2013 05:15 AM
Functional equation: x + 2*y + z = x @ (y @ z) antros Applied Math 4 October 12th, 2008 12:12 PM
another functional equation desum Elementary Math 0 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.