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a-b-gt-0-and-x-y-R-find-all-functions-f-R-R-such-that-f-ax-f-by-abf-bx-ay-x-a-y-b-f-x-b-y-a-




Question Number 140587 by mathsuji last updated on 09/May/21
a;b>0 and ∀x;y∈R  find all functions f:R→R ; such that  f(ax)f(by)=abf(bx+ay)+x^a y^b f(x^b y^a )
$${a};{b}>\mathrm{0}\:{and}\:\forall{x};{y}\in\mathbb{R} \\ $$$${find}\:{all}\:{functions}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:;\:{such}\:{that} \\ $$$${f}\left({ax}\right){f}\left({by}\right)={abf}\left({bx}+{ay}\right)+{x}^{{a}} {y}^{{b}} {f}\left({x}^{{b}} {y}^{{a}} \right) \\ $$

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