Stability of a functional equation for square root spirals

Keywords---Hyers-Ulam-Rassias stability, Functional equation, Riemann zeta function, Square root spital. 1. I N T R O D U C T I O N The staxting point of studying the stability of functional equations seems to be the famous talk of Ulam [2] in 1940, in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms. Let G1 be a group and let G2 be a metric group with a metric d(., .). Given e > 0, does there exist a ~ > 0 such that if a mapping h : GI --, G2 satisfies the inequality d(h(xy), h(x)h(y)) < ~ for all x, y E G1, then there exists a homomorphism H : G 1 --* G2 with d(h(x),H(x)) < 6 for all x E GI? The ca.se of approximately additive mappings was solved by Hyers [3] under the assumption that G1 and G2 are Banach spaces. Later, the result of Hyers was significantly generalized by Rassias [4] (see also [5]). It should be remarked that we can find in [6,7] a lot of references concerning the stability of functional equations (see also [8-10]). Recently, Heuvers, Moak and Boursaw investigated the general solution of the functional equation f = f(r) + &rctan r ' (1) 0893-9659/02/$ see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. Typeset by .A~B-TEX PII: S0893-9659(01)00155-0 436 S.-M. JUNG AND P. K. SAHOO « which is closély related to the square root spiral, for the Gase that f (1) = 0 and f ( r ) i s monotone increasing for r > 0 (see [1]). In this note, we follow the method of Heuvers, Moak and Boursaw [3] to prove a Hyers-UlamRassias stability (or a general type of Hyers-Ulam stability) of the functional equation (1). The main result of this note is presented in the following theorem. THEOREM 1. I[a mapping f : [1, oo) ---, [0, oo) satisfies the inequality [f ( ~ ) f ( r ) a r c t a n l l <_ @(r), (2) for all r E [1, oo), where ~ : [1, c~) --, [0, oo) is a mapping which satisfies the condi¢ion