Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


New generalizations of Jensen’s functional equation
HTML articles powered by AMS MathViewer

by Hiroshi Haruki and Themistocles M. Rassias PDF
Proc. Amer. Math. Soc. 123 (1995), 495-503 Request permission


Let f be an unknown entire function of a complex variable, and let s, t be real variables. We consider Jensen’s functional equation \[ f\left ( {\frac {{x + y}}{2}} \right ) = \frac {{f(x) + f(y)}}{2},\] 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 \[ \left | {f\left ( {\frac {{s + it}}{2}} \right )} \right | = \left | {\frac {{f(s) + f(it)}}{2}} \right |.\] The main purpose of this paper is to solve a new generalization of the above equation.
  • J. Aczél, Lectures on functional equations and their applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser. MR 0208210
  • J. Aczél and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. With applications to mathematics, information theory and to the natural and social sciences. MR 1004465, DOI 10.1017/CBO9781139086578
  • J. Aczél and E. Vincze, Über eine gemeinsame Verallgemeinerung zweier Funktionalgleichungen von Jensen, Publ. Math. Debrecen 10 (1963), 326–344 (German). MR 166507
  • Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
  • Boo Rim Choe, A functional equation of Pexider type, Funkcial. Ekvac. 35 (1992), no. 2, 255–259. MR 1189895
  • Hiroshi Haruki, On the equivalence of Hille’s and Robinson’s functional equations, Ann. Polon. Math. 28 (1973), 261–264. MR 342685, DOI 10.4064/ap-28-3-261-264
  • Hiroshi Haruki, On a functional equation of Pexider type, Aequationes Math. 36 (1988), no. 1, 1–19. MR 959790, DOI 10.1007/BF01837968
  • Hiroshi Haruki, A new quadratic equation, Constantin Carathéodory: an international tribute, Vol. I, II, World Sci. Publ., Teaneck, NJ, 1991, pp. 476–488. MR 1130850
  • Hiroshi Haruki, A new cosine functional equation, The mathematical heritage of C. F. Gauss, World Sci. Publ., River Edge, NJ, 1991, pp. 334–341. MR 1146237
  • Einar Hille, A Pythagorean functional equation, Ann. of Math. (2) 24 (1922), no. 2, 175–180. MR 1502636, DOI 10.2307/1967713
  • —, A class of functional equations, Ann. of Math. (2) 29 (1928), 215-222. C. T. Ng, The Jensen equation on groups, Aequationes Math. 37 (1989).
  • R. M. Robinson, A curious trigonometric identity, Amer. Math. Monthly 64 (1957), 83–85. MR 82549, DOI 10.2307/2310381
  • Problems and solutions section, Amer. Math. Monthly 99 (1992), 875.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 39B32, 30D05
  • Retrieve articles in all journals with MSC: 39B32, 30D05
Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 495-503
  • MSC: Primary 39B32; Secondary 30D05
  • DOI:
  • MathSciNet review: 1224617