Bounds of certain harmonic sums
For fixed integers \(p,q,r\) and \(s\) with \(0<p<q\) and \(0\leq p+r< q+s\), define the constant \(\lambda= (ps-qr)/(q-p)\) and \[ \Phi(n) =(pn+ r+1)^{-1} +(pn+ r+2)^{-1}+ \cdots +(qn +s)^{-1} \] for each integer \(n\geq 1\). The main result shows that \(\Phi(n)\) is an increasing function of \(n\) if \(\lambda\leq {1\over 2}\), that \(\Phi(n)\) is a decreasing function if \(\lambda \geq 1\), and that \(\Phi(n)\) is increasing on \([0,n_0]\) and decreasing on \([n_0,\infty)\), for some \(n_0\geq 1\), if \(\lambda\in ({1\over 2},1)\). This result leads to upper and lower bounds for \(\Phi\), and extends the work of \textit{D. D. Adamović} and \textit{M. R. Tasković} [Publ. Fac. Electrotechn. Univ. Belgrade, Sér. Math. Phys. 247-273, 41-50 (1969; Zbl 0183.05001)], \textit{J. E. Pečarić} [Mat. Bilt. 5/6, 29-36 (1982; Zbl 0592.26018)] and \textit{S. Simić} [Mat. Vesn. 38, 331-336 (1986; Zbl 0648.40002)] in the cases where \(r=0=s\); \(r=0\), \(s=1\); and \(r=-1=s\).
scientific article; zbMATH DE number 1003950