J. oflnequal. & Appl., 1998, Vol. 2, pp. 373-380 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint. Printed in India. On Some Inequalities and Stability Results Related to the Exponential Function CLAUDI ALSINA a,, and ROMAN GER b a Sec. Matemtiques Informtica, Univ. Polit+cnica de Catalunya, Diagonal 649, 08028 Barcelona, Spain; blnstitute of Mathematics, Silesian University, ul. Bankowa 14, 40-007 Katowice, Poland (Received 12 November 1997, Revised 20 February 1998) Some inequalities related to the exponential function are solved and the stability of the functional equationsf’(x)-f(x) and (f(y)-f(x))/(y-x)=f((x + y)/2) is studied. Keywords." Inequalities; Exponential function; Hyers-Ulam stability; Functional equation AMS 1991 Subject Classification." 39C05 One of the most classical characterizations of the real exponential function f(x)- e is the fact that the exponential function is the only (modulo a multiplicative constant) nontrivial solution of the differential equation f’=f Our aim in this note is to study the Hyers-Ulam stability of this equation, i.e. to solve for a given c > 0 the inequality }f’(x)-f(x)l _< e, (1) and to study also the related inequality (for all x =/= y) (2) Corresponding author. E-mail: alsina@ea.upc.es 373