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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.

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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