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On a functional equation connected with Rao's quadratic entropy

Authors: J. K. Chung, B. R. Ebanks, C. T. Ng and P. K. Sahoo
Journal: Proc. Amer. Math. Soc. 120 (1994), 843-848
MSC: Primary 39B22
MathSciNet review: 1180464
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Abstract: We determine the general solution of the functional equation fxy,

$\displaystyle f\left( {\frac{{x + y}} {2}} \right) + f\left( {\frac{{x - y}} {2... ...t( {\frac{x} {2}} \right) + 2f\left( {\frac{y} {2}} \right) + \lambda f(x)f(y),$

/: [- $ f:[ - 1,1] \to {\mathbf{R}}$. This equation was used by Lau in order to characterize Rao's quadratic entropies. The general solution is obtained here as a special case of a more general result for $ f$ mapping a neighborhood of 0 in linear topological space into a field.

References [Enhancements On Off] (What's this?)

  • [1] János Aczél, The general solution of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 20 (1965), 65–73 (English, with Serbo-Croatian summary). MR 198023
  • [2] J. K. Chung, B. R. Ebanks, C. T. Ng, and P. K. Sahoo, On a quadratic-trigonometric functional equation and some applications, submitted.
  • [3] Ka-Sing Lau, Characterization of Rao’s quadratic entropies, Sankhyā Ser. A 47 (1985), no. 3, 295–309. MR 863724
  • [4] László Székelyhidi, Convolution type functional equations on topological abelian groups, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. MR 1113488

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Keywords: Biadditive function, quadratic functional equation, quadratic entropy
Article copyright: © Copyright 1994 American Mathematical Society