Generalized double-integral Ostrowski type inequalities on time scales (Q540315)

The inequality proved by A. M. Ostrowski in 1938 asserts that if \(f:[a,b]\to\mathbb R\) is a continuous function whose derivative on \((a,b)\) satisfies \(|f'|\leq M\), then \[ \left| f(x)-\frac{1}{b-a}\int_a^b f(t)\,dt\right|\leq \left(\frac{1}{4}+\left(\frac{x-\frac{a+b}{2}}{b-a}\right)^2\right)(b-a)M \] for every \(x\in[a,b]\). A time-scale version of Ostrowski's inequality was later obtained by \textit{M. Bohner} and \textit{T. Matthews} [JIPAM, J. Inequal. Pure Appl. Math. 9, No.~1, Paper No.~6 (2008; Zbl 1178.26020)]. In the subsequent period, several authors have investigated Ostrowski-type inequalities for time-scale functions of two variables (see the references in the paper). The main result of the present paper is the following theorem: Consider a pair of time scales \(\mathbb T_1\), \(\mathbb T_2\) and a function \(f:[a,b]_{\mathbb T_1}\times [c,d]_{\mathbb T_2}\to\mathbb R\) whose partial derivatives \[ \frac{\partial f}{\Delta_2 s}(t,s), \quad\frac{\partial f}{\Delta_1 t}(t,s),\;\text{ and }\;\frac{\partial^2 f}{\Delta_2 s\Delta_1 t} \] are continuous in \([a,b]_{\mathbb T_1}\times [c,d]_{\mathbb T_2}\) and their absolute values are bounded by \(M_1\), \(M_2\), and \(M_3\). Then for every \(k\in\mathbb N\), \(x\in[a,b]_{\mathbb T_1}\), \(y\in[c,d]_{\mathbb T_2}\), we have \[ \left|f(x,y)-\frac{1}{(h_k(x,a)-h_k(x,b))(h_k(y,c)-h_k(y,d))}\int_a^b\int_c^d \Phi(x,t)\Phi(y,s)f(\sigma_1(t),\sigma_2(s))\Delta_2s\Delta_1t\right| \] \[ \begin{multlined} \leq \frac{1}{(h_k(x,a)-h_k(x,b))(h_k(y,c)-h_k(y,d))}\\ \big[M_1(h_k(x,a)+(-1)^k h_k(x,b))(h_{k+1}(y,c)+(-1)^{k+1}h_{k+1}(y,d)) +h_{k+1}(x,c)\\ +(-1)^{k+1}h_{k+1}(x,d))(M_2(h_k(y,a)+(-1)^k h_h(y,b))\\ +M_3(h_{k+1}(y,c)+(-1)^{k+1}h_{k+1}(y,d)))\big],\end{multlined} \] where \(h_k\) are the generalized polynomial functions corresponding either to \(\mathbb T_1\) or \(\mathbb T_2\) (the right time scale has to be chosen depending on the context). The meaning of \(\Phi\) is not explained in the statement of the main theorem, but it appears later in the paper: Given a time scale \(\mathbb T\), we have \[ \Phi(t,s)=\begin{cases} h_{k-1}(s,a),& s\in[a,t)_{\mathbb T},\\ h_{k-1}(s,b),& s\in[t,b]_{\mathbb T}\end{cases} \] for every \(k\in\mathbb N\) and \(s\), \(t\in\mathbb T\). After proving the main result, the authors explain that several Ostrowski-type inequalities obtained in earlier papers are simple corollaries of their theorem. Finally, they discuss the special cases \(\mathbb T_1=\mathbb T_2=\mathbb R\), \(\mathbb T_1=\mathbb T_2=\mathbb Z\), and \(\mathbb T_i=q_i^{\mathbb N_0}\) with \(q_1\), \(q_2>1\).