a tough functional equation

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#1253

From: Paul Bruckman
Date: Wed, 29 Jul 1998 00:00

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I need help in solving (if possible) the following functional equation
for M(t), where t is any real positive number :

M(sinh u) = K(u) + G(u)*M(K(u)/G(u)) , where (if x = exp(-u)),

K(u) = (1-x)/2 , G(u) = (1-x^2)((x^2+1)^(3/2) + x^3 - 1)/(4x^2).

We are also given some boundary conditions :  M(0) = 0 ,
M(1/Sqrt(3)) = 1/3 ; letting u-->oo, we obtain :
M(oo) = 1/2 + 3/8 * M(4/3).  I have obtained the approximate values of
M(t) for about 20 different values of t, mostly in the interval (0,1].
I am looking for the general formula for M(t).

The above problem occurs within the context of the limit point of
successive Malfatti circles constructed inside a given initial triangle.
However, I am posing it merely as an exercise in functional analysis. 

All help will be appreciated.  Sincerely, Paul Bruckman

[ Moderator's note: a closed form solution seems to me rather unlikely,
but maybe analysis can supply some useful information - R.I. ]

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From: Paul Bruckman <aardwolf@gte.net>
Subject: a tough functional equation
Date: 1998/07/29
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