General Mathematics Vol. 15, No. 1 (2007), 75–80 About the use of a result of Professor Alexandru Lupa¸ s to obtain some properties in the theory of the number e 1 Andrei Vernescu Dedicated to Professor Alexandru Lupa¸ s on his 65th anniversary Abstract A very elegant result of Professor Alexandru Lupa¸ s gives us that the point c“ of the mean value theorem (of Lagrange) applied to the logarithmic function on an interval [a, b] ⊂ (0, ∞) has the property that √ ab < c < (a + b)/2. In this paper we show that the result of Professor Alexandru Lupa¸ s is decisively useful to establish some other nice results in the theory of the number e and in some related question. 2000 Mathematics Subject Classification: 26D07, 26D15. Key words and phrases: The arithmetic, geometric and harmonic means, the number e, the logarithm. 1. The work of Professor Alexandru Lupa¸ s contains numerous, var- ious and elegant results especially in the following domains: Real func- tions, Inequalities, Sequences and Series, Special functions, Numerical Anal- ysis, Umbral Calculus, Approximation Theory, but also Computer Sciences, Computational Mathematics and Experimental Mathematics. 1 Received 1 February, 2007 Accepted for publication (in revised form) 13 March, 2007 75