J ournal of Mathematical I nequalities Volume 5, Number 2 (2011), 199–211 NOTE ON AN INEQUALITY OF GAUSS J OSIP PE ˇ CARI ´ C AND KSENIJA SMOLJAK Abstract. In this paper a functional defined as the difference between the left-hand and the right- hand side of an extension of the Gauss inequality given in [H. Alzer, On an inequality of Gauss, Rev. Mat. Complut. 4(2) (1991), 179–183.] is studied. Related analogous of the Lagrange and the Cauchy mean value theorems are obtained. Furthermore, Gauss means are generated and their monotonicity property is proven. 1. Introduction Let us recall the inequality of Gauss (see, [8, p. 195]): Let f : [0, ∞) → R be a decreasing function, then, for all real numbers k > 0, k 2 ∞  k f (x)dx  4 9 ∞  0 x 2 f (x)dx. (1.1) H. Alzer proved in 1991 (see [1]) that an application of the following theorem leads to a new proof and to a converse of inequality (1.1) : THEOREM 1.1. Let g : [a, b] → R be strictly increasing, convex and differentiable, and let f : I → R be decreasing. Then b  a f (s(x))g ′ (x)dx  g(b)  g(a) f (x)dx  b  a f (t (x))g ′ (x)dx, (1.2) where s(x)= g(b) - g(a) b - a (x - a)+ g(a) (1.3) and t (x)= g ′ (x 0 )(x - x 0 )+ g(x 0 ), x 0 ∈ [a, b]. (1.4) (I ⊆ R is an interval containing a, b, g(a), g(b), t (a) and t (b).) If either g is convave (instead of convex) or f is increasing, then the reversed inequalities hold. Mathematics subject classification (2010): Primary 26D10, Secondary 26D15. Keywords and phrases: Gauss inequality, mean value theorems, exponential convexity, means. c  , Zagreb Paper JMI-05-18 199