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New generalizations of Jensen's functional equation

Authors: Hiroshi Haruki and Themistocles M. Rassias
Journal: Proc. Amer. Math. Soc. 123 (1995), 495-503
MSC: Primary 39B32; Secondary 30D05
MathSciNet review: 1224617
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Abstract: Let f be an unknown entire function of a complex variable, and let s, t be real variables. We consider Jensen's functional equation

$\displaystyle f\left( {\frac{{x + y}}{2}} \right) = \frac{{f(x) + f(y)}}{2},$

where x, y are complex variables. Replacing x and y by s and it in the above equation and taking the absolute values of the resulting equality one obtains the functional equation

$\displaystyle \left\vert {f\left( {\frac{{s + it}}{2}} \right)} \right\vert = \left\vert {\frac{{f(s) + f(it)}}{2}} \right\vert.$

The main purpose of this paper is to solve a new generalization of the above equation.

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

  • [1] J. Aczél, Lectures on functional equations and their applications, Academic Press, New York and London, 1966. MR 0208210 (34:8020)
  • [2] J. Aczél and J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, Cambridge, New York, New Rochelle, Melbourne, and Sydney, 1989. MR 1004465 (90h:39001)
  • [3] J. Aczél and E. Vincze, Über eine gemeinsame Verallgemeinerung zweier Funktional-gleichungen von Jensen, Publ. Math. Debrecen 10 (1963), 326-344. MR 0166507 (29:3782)
  • [4] L. V. Ahlfors, Complex analysis, 2nd ed., McGraw-Hill, New York, 1966. MR 510197 (80c:30001)
  • [5] Boo Rim Choe, A functional equation of Pexider type, Funkcial. Ekvac. 35 (1992), 255-259. MR 1189895 (94b:39033)
  • [6] H. Haruki, On the equivalence of Hille's and Robinson's functional equations, Ann. Polon. Math. 28 (1973), 261-264. MR 0342685 (49:7430)
  • [7] -, On a functional equation of Pexider type, Aequationes Math. 36 (1988), 1-19. MR 959790 (89k:39017)
  • [8] -, A new quadratic equation, Constantin Caratheódory: An International Tribute (Th. M. Rassias, ed.), World Scientific, Singapore, New Jersey, and London, 1991, pp. 476-488. MR 1130850 (92h:39013)
  • [9] -, A new cosine functional equation, The Mathematical Heritage of C. F. Gauss (G. M. Rassias, ed.), World Scientific, Singapore, New Jersey, and London, 1991, pp. 334-341. MR 1146237 (93a:39018)
  • [10] E. Hille, A Pythagorean functional equation, Ann. of Math. (2) 24 (1923), 175-180. MR 1502636
  • [11] -, A class of functional equations, Ann. of Math. (2) 29 (1928), 215-222.
  • [12] C. T. Ng, The Jensen equation on groups, Aequationes Math. 37 (1989).
  • [13] R. M. Robinson, A curious trigonometric identity, Amer. Math. Monthly 64 (1957), 83-85. MR 0082549 (18:568f)
  • [14] Problems and solutions section, Amer. Math. Monthly 99 (1992), 875.

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Keywords: Unknown entire function, Jensen's functional equation, Cosine functional equation, Robinson's functional equation, Hille's functional equation
Article copyright: © Copyright 1995 American Mathematical Society

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