ON CERTAIN SEQUENCES DERIVED FROM GENERALIZED EULER-MASCHERONI CONSTANTS

Let 0 < a < 1, and let C(alpha) := lim(n ->infinity) (1 + 1/2(alpha) + ... + 1/n(alpha) - n(1-alpha)/1-alpha). It is proved that there exists a unique sequence (omega(n)) such that 1 + 1/2(alpha) + ... + 1/n(alpha) - C(alpha) + (n + omega(n))(1-alpha)/1-alpha. Moreover, the sequence (omega(n)) is decreasing and satisfies 1/2 <= omega(n) <= 1/4 [1+ (1 + 1/n](alpha)], whence lim(n ->infinity) omega(n) = 1/2. This is only a special case of the more general results established in this paper. These results concern some sequences derived from generalized Euler-Mascheroni constants involving convex functions and complement similar ones obtained by V. Timofte [Integral estimates for convergent positive series. J. Math. Anal. Appl. 303 (2005), 90-102].