Sharpening Hölder's inequality (Q5917768)

In this paper, we show that sharpening Cauchy's inequality, proved by \textit{J. W. Hovenier} [J. Math. Anal. Appl. 186, No. 1, 156-160 (1994; Zbl 0815.26012); Erratum, ibid. 190, No. 1, 294 (1995; Zbl 0823.26014)], has its generalization in the case of Hölder's inequality [\textit{D. S. Mitrinovíc}, \textit{J. E. Pečarić} and \textit{A. M. Fink}: ``Classical and new inequalities in analysis'' (1993; Zbl 0771.26009), p. 105].