Haruki, H

Rassias, T

Hiroshi Haruki

Themistocles M. Rassias

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Proceedings of the American Mathematical Society

New generalizations of Jensen’s functional equation

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Previous Up NextCitationsFrom References: 4From Reviews: 0MR1224617 (95c:39019) 39B32 30D05Haruki, Hiroshi (3-WTRLM-PM) ; Rassias, Themistocles M. (GR-LAV)New generalizations of Jensen’s functional equation. (English summary)Proc. Amer. Math. Soc. 123 (1995), no. 2, 495–503.While the formulation of the paper’s results is quite complicated, it can be simplified tothe following. The functional equation|f((s+ it)/2)|r = |f(s) + f(it)|/2,where s and t are real variables and r a positive rational constant, has nonconstantentire solutions only for r = 1 (f(z) = Az+B) and r = 2 (as determined previously byHaruki [in The mathematical heritage of C. F. Gauss, 334–341, World Sci. Publishing,River Edge, NJ, 1991; MR1146237]: f(z) = eic cos(bz), where c is a real and b a complexconstant). A nine-page, rather technical proof is offered. J. Acze´lReferences1. J. Acze´l, Lectures on functional equations and their applications, Academic Press,New York and London, 1966. MR02082102. J. Acze´l and J. Dhombres, Functional equations in several variables, CambridgeUniv. Press, Cambridge, New York, New Rochelle, Melbourne, and Sydney, 1989.MR10044653. J. Acze´l and E. Vincze, U¨ber eine gemeinsame Verallgemeinerung zweier Funktional-gleichungen von Jensen, Publ. Math. Debrecen 10 (1963), 326-344. MR01665074. L. V. Ahlfors, Complex analysis, 2nd ed., McGraw-Hill, New York, 1966. MR05101975. Boo Rim Choe, A functional equation of Pexider type, Funkcial. Ekvac. 35 (1992),255-259. MR11898956. H. Haruki, On the equivalence of Hille’s and Robinson’s functional equations, Ann.Polon. Math. 28 (1973), 261-264. MR03426857. H. Haruki, On a functional equation of Pexider type, Aequationes Math. 36 (1988),1-19. MR09597908. H. Haruki, A new quadratic equation, Constantin Caratheo´dory: An InternationalTribute (Th. M. Rassias, ed.), World Scientific, Singapore, New Jersey, and London,1991, pp. 476-488. MR11308509. H. Haruki, 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. MR114623710. E. Hille, A Pythagorean functional equation, Ann. of Math. (2) 24 (1923), 175-180.MR150263611. E. Hille, A class of functional equations, Ann. of Math. (2) 29 (1928), 215-222.MR150283312. 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. MR008254914. Problems and solutions section, Amer. Math. Monthly 99 (1992), 875. MR1542215Note: This list reflects references listed in the original paper as accurately aspossible with no attempt to correct errors.c© Copyright American Mathematical Society 1995, 2016

New generalizations of Jensen’s functional equation

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