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This question pertains to "A Entropy view of Fibonacci Trees" (1982) by Yasuichi Horibe, Zbl 0491.94009.
Horibe defines a binary tree where a node has a left branch with probability x and a right branch with probability $1-x$. The left branch has cost 1 and the right branch has cost c. In equation (4), p. 171 of the paper, the entropy of a node per cost is provided as $$ { H(x, 1-x)} \over{x + c (1-x)} $$ Horibe states "Let $\lambda$ be the maximizing value of x. By differentiating, $\lambda$ is the unique positive root of $x^c= 1 - x$."

I am afraid I am unable to see how this is found. Can anyone on MO show me? Feeding the question to Mathematica or GPT4 produces no useful answer. Nor do I as yet follow the next sentence "The maximum value of the function is $-\log \lambda$."

Thank you,

Bill

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    ${ H(x, 1-x)} \over{x + c (1-x)}$ with $H(x,1-x)=-x\log x-(1-x)\log(1-x)$ has the derivative $\log[(1-x)/x^c]/(c+x-cx)^2$ which vanishes when $(1-x)/x^c=1$; at that value of $x$ a simple substitution of $c=\log(1-x)/\log x$ immediately gives $\frac{H(x, 1-x)} {x + c (1-x)}=-\log x$. – Carlo Beenakker Feb 20 '24 at 21:04

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