MathDB

Let

f : R^+ \to R^+

be a function satisfying

f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}

for all positive real numbers

x_1, x_2

. Show that

f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}

for all positive integers

n

and positive real numbers

x_1, x_2,..., x_n

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