Vol. 26, No. 1, 1968

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ISSN: 0030-8730
On inequalities generalizing a Pythagorean functional equation and Jensen’s functional equation

Hiroshi Haruki

Vol. 26 (1968), No. 1, 85–90
Abstract

Several authors have solved the Pythagorean functional equation

|f(x ⊢ iy)|2 = |f(x)|2 + |f(iy)|2,
(1)

where f is an entire function and x and y are real variables.

A simple computation shows that, if f is a solution of (1), then f is also a solution of

                                -            -
|f(z1 +z2)|2 + |f(z1 − z2)|2 = |f(z1 + z2)|2 + |f(z1 − z2)|2,
(2)

where z1 and z2 are complex variables. (If an entire function vanishes at the origin and is a solution of (2), then it is a solution of (1), and conversely.) If an entire function f is a solution of Jensen’s functional equation

f(z1 + z2)+ f(z1 − z2) = 2f(z1),
(3)

where z1 and z2 are complex variables, then it is also a solution of

|f(z + z )+ f(z − z )| = |f(z + z )+ f(z − z )|.
1   2      1   2       1   2      1   2
(4)

In this paper we shall prove that a solution of (4) is always a solution of (2). Then we shall solve certain functional inequalities derived from (2) and use the solutions to solve (1), (2), (3), and (4).

Mathematical Subject Classification
Primary: 39.30
Milestones
Received: 19 January 1967
Published: 1 July 1968
Authors
Hiroshi Haruki