The authors note that the inequality
does not always hold for (in particular not always for ) under the assumptions stated in “Analytic inequalities” (1970; Zbl 0199.38101), pp. 58–59, of D. S. Mitrinović. (Note: This error has been noticed also by M. Bjelica [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 19, 105–109 (1990; Zbl 0733.26011)], and by L. Losonczi and Zs. Páles [J. Math. Anal. Appl. 205, No. 1, 148–156 (1997; Zbl 0871.26012)]. The latter also generalized the corrected inequality.
The present authors offer among others the inequality
under the above assumptions, and generalizations.