Note on a discrete Opial-type inequality (Q1901992)

For any sequence \((u_i)\) of real numbers, the sequences \((\Delta^k u_i)\) \((k= 0, 1,\dots)\) are defined recursively by \(\Delta^0 u_i= u_i\), \(\Delta^k u_i= \Delta^{k- 1} u_{i+ 1}- \Delta^{k- 1} u_i\) \((k= 1,2,\dots)\). In the present note, the author proves that if \(p_i\geq 0\) \((i= 0,1,\dots, k- 1)\), \(p_k> 0\) are real numbers satisfying \(p_0+\cdots+ p_{k- 1}> 0\) and \(p= p_0+\cdots+ p_{k- 1}+ p_k> 1\); \(a_v\geq 0\), \(b_v> 0\) \((v= 0, 1,\dots, n- 1)\) are real numbers; and \((u_i)\), \((\Delta u_i),\dots, (\Delta^k u_i)\) are nonnegative sequences with \(u_0= \Delta u_0=\cdots= \Delta^{k- 1} u_0= 0\), then \[ \sum^{n- 1}_{v= 0} \Biggl(a_v \prod^k_{i= 0} (\Delta^i u_v)^{p_i}\Biggr)\leq c^\alpha_n \prod^{k- 1}_{i= 0} [(k- i- 1)!]^{- p_i} S_n(a,b)\cdot \sum^{n- 1}_{v= 0} b_v(\Delta^k u_v)^p, \] where \(\alpha= p_k/p\), \(c_n\) is given by \[ c_1= 0,\;c_v= \alpha(1- \alpha)^{(1- \alpha)/\alpha}[1- c_{v- 1}]^{(\alpha- 1)/\alpha}\qquad (v= 2,\dots, n), \] and \[ S_n(a, b)= \Biggl[\sum^{n- 1}_{v= 0} a_v^{p/(p- p_k)} b_v^{- p_k/(p- p_k)}\cdot \prod^{k- 1}_{i= 0} B_{iv}^{p_i(p- 1)/(p- p_k)}\Biggr]^{(p- p_k)/p}, \] \[ B_{iv}= \sum^{v- 1}_{j=0} ((v- j- 1)^{(k- i- 1)p/(p- 1)} b_j^{- 1/(p- 1)}). \] Since \((c_n)\) is strictly increasing and \(c_n\) tends to \(\alpha\) as \(n\to \infty\), the last discrete inequality is an improvement of a known result due to \textit{J.-D. Li} [J. Math. Anal. Appl. 167, No. 1, 98-110 (1992; Zbl 0821.26014)].