On generalized Euler constants and Schlömilch-Lemonnier type inequalities (Q864648)

Following a suggestion of Boas the author considers the two sequences \(A_n = \sum_{i=1}^nF(i) -\int_1^{n+1}F\) and \(B_n= \sum_{i=1}^{n+1}F(i) -\int_1^{n+1}F\) where \(F:[1,\infty[\mapsto ]0, \infty[\) is strictly decreasing and continuous. It is proved that both of the sequences converge the first being strictly increasing the second strictly decreasing. Further if \(\lim_{x\to \infty} F(x) = 0\) the two limits are equal, \(\gamma_F\) say . The case \(F(x) = 1/x\) is classical and \(\gamma_F\) is Euler's constant \( \gamma\). In this case it is known that \(\lim_{n\to \infty} n(B_n- \gamma) =1/2\), [\textit{R. M. Young}, Math. Gaz. 75(472), 187--190 (1991)], and the author extends this to the general case when \(F\) is strictly convex and \(G(x)= xF(x)\) is concave, to obtain \(\lim_{n\to \infty} {1\over F(n)}(B_n- \gamma) =1/2\). Further results are obtained by considering multiplicative analogues of the above sequences.