Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON A FUNCTIONAL EQUATION ARISING FROM PROTH IDENTITY

  • Received : 2015.06.08
  • Published : 2016.01.31
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Abstract

We determine the general solutions $f:\mathbb{R}^2{\rightarrow}\mathbb{R}$ of the functional equation f(ux-vy, uy+v(x+y)) = f(x, y)f(u, v) for all x, y, u, $v{\in}\mathbb{R}$. We also investigate both bounded and unbounded solutions of the functional inequality ${\mid}f(ux-vy,uy+v(x+y))-f(x,y)f(u,v){\mid}{\leq}{\phi}(u,v)$ for all x, y, u, $v{\in}\mathbb{R}$, where ${\ph}:\mathbb{R}^2{\rightarrow}\mathbb{R}_+$ is a given function.

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References

  1. M. Albert and J. A. Baker, Bounded solutions of a functional inequality, Canad. Math. Bull. 25 (1982), no. 4, 491-495. https://doi.org/10.4153/CMB-1982-071-9
  2. R. Blecksmith and S. Broudno, Equal sums of three fourth powers or what Ramanujan could have said, Math. Magazine 79 (2006), 297-301. https://doi.org/10.2307/27642955
  3. E. A. Chavez and P. K. Sahoo, On a functional equation arising from number theory, Appl. Math. Lett. 24 (2011), no. 3, 344-347. https://doi.org/10.1016/j.aml.2010.10.020
  4. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, 1998.
  5. P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Taylor & Francis Group, Boca Raton, 2011.