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Quadratic functions fulfilling an additional condition along hyperbolas or the unit circle

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Let \( S_1 \) denote the set of all pairs (xy) of real numbers that fulfill the condition \( x^2 - y^2 = 1 \), and \( S_2 \) denote the set of all pairs (xy) of real numbers that fulfill the condition \( x^2 + y^2 = 1 \,\). In this paper we consider quadratic real functions f that satisfy the additional equation \( y^2 f(x) = x^2 f(y) \) under the condition \( (x,y) \in S_i \)\((i=1,2)\). We prove that each of these conditions implies \( f(x) = f(1) x^2 \) for all \( x \in \mathbb {R} \).

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The author would like to thank Zoltán Boros for helpful suggestions and his constant support and also the anonymous referee for his/her essential comments and useful advice.

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Correspondence to Edit Garda-Mátyás.

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Garda-Mátyás, E. Quadratic functions fulfilling an additional condition along hyperbolas or the unit circle. Aequat. Math. 93, 451–465 (2019).

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