Quadratic functions satisfying an additional equation
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QUADRATIC FUNCTIONS SATISFYING AN ADDITIONAL EQUATION M. AMOU Division of Pure and Applied Science, Graduate School of Science and Technology, Gunma University, Tenjin-cho 1-5-1, Kiryu 376-8515, Japan e-mail: [email protected] (Received July 25, 2019; revised March 7, 2020; accepted March 13, 2020)
Abstract. There is a result, due independently to Kurepa [14] and to Jurkat [12], which distinguishes linear functions or derivations from other additive functions as solutions to certain functional equations. The purpose of this paper is to prove an analogue of a part of this result, corresponding to derivations, for quadratic functions.
1. Introduction Throughout this paper we denote by R the field of real numbers, and by Q the field of rational numbers. Let us set R× := R\{0}. A function ϕ : R → R is called additive if it satisfies ϕ(x + y) = ϕ(x) + ϕ(y),
x, y ∈ R.
It is known that an additive function ϕ satisfies ϕ(rx) = rϕ(x) for x ∈ R and for r ∈ Q (see [13, Theorem 5.2.1]). Therefore, ϕ is a Q-linear map on R. An additive function ϕ : R → R is called a linear function if ϕ(x) = ϕ(1)x for x ∈ R, and a derivation if it satisfies ϕ(xy) = xϕ(y) + yϕ(x),
x, y ∈ R.
It is easily seen that, if ϕ is a linear function or a derivation, then ϕ satisfies 1 , x ∈ R× , (1.1) ϕ(x) = δx2 ϕ x where δ = 1 or δ = −1, respectively. Conversely, answering a question posed by Halperin [10] (see also Acz´el [1]), the following result was established Key words and phrases: quadratic function, derivation, conditional equation. Mathematics Subject Classification: 39B22, 39B55. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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M. M. AMOU AMOU
by Kurepa [14], and independently by Jurkat [12] (see also [13, Theorem 14.3.3]). Theorem 1.1 [12,14]. Let ϕ : R → R be an additive function satisfying (1.1). Then ϕ is a linear function if δ = 1; while ϕ is a derivation if δ = −1. In connection with this result, for given δ ∈ {1, −1}, it seems natural to consider quadratic functions f which satisfy 1 (1.2) f (x) = δx4 f , x ∈ R× . x Here a function f : R → R is called quadratic if it satisfies (1.3)
f (x + y) + f (x − y) = 2f (x) + 2f (y),
x, y ∈ R.
For example, if ϕ, ψ : R → R are additive functions, then ϕ(x)ψ(x) and ϕ(x2) are quadratic functions (see the beginning of Section 3 for more about quadratic functions). In the case where δ = 1, Grz¸a´slewicz [8,9] studied the equation (1.2) under certain additional conditions. Very recently, in the same case, GardaM´aty´as [7] found remarkable properties which should be fulfilled by any quadratic solution to the equation (1.2) (see Remark 3.1). However, it seems that there remains still open to determine all the quadratic solutions to (1.2) with δ = 1. On the other hand, in the case where δ = −1, although it is easily seen that every quadratic function f of the form f (x) = xϕ(x) with a derivation ϕ is a solution to the equation (1.2), it seems that any other solution besides this example has not been found. This
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