New Ostrowski-type inequalities and their applications in two coordinates. (English) Zbl 1349.26026
In this paper, the author derives Ostrowski-type inequalities in two coordinates using the classical Ostrowski inequality. He gets in Theorems 2.1, 2.3 and 2.5 the inequalities
\[ \begin{aligned} &\left | \int \limits_{a}^{b}\frac {f\left (x,c\right) +f\left (x,d\right)}{2}{\operatorname {d}}x+\int \limits_{c}^{d}\frac {f\left (a,y\right) +f\left (b,y\right)}{2}{\operatorname {d}}x\right . \\ &\left . -\left (\frac {1}{b-a}+\frac {1}{d-c}\right)\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right) \\ \leq {} &\frac {M+N}{2}\left (b-a\right) \left (d-c\right), \end{aligned} \] \[ \begin{aligned} &\left | \frac {1}{4}\left [\frac {1}{b-a}\int \limits_{a}^{b}\bigl (f\left (x,c\right) +f\left (x,d\right)\bigr) {\operatorname {d}}x+\frac {1}{d-c}\int \limits_{c}^{d}\bigl (f\left (a,y\right) + f\left (b,y\right)\bigr) {\operatorname {d}}y\right]\right . \\ &\left . -\frac {1}{\left (b-a\right) \left (d-c\right)}\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {1}{4}\bigl (\left (b-a\right) M+\left (d-c\right) N\bigr), \end{aligned} \] and \[ \begin{aligned} &\left | \int \limits_{a}^{b}f\left (x,\frac {c+d}{2}\right)dx+\int \limits_{c}^{d}f\left (\frac {a+b}{2},y\right) {\operatorname {d}}y\right . \\ &\left . -\left (\frac {1}{b-a}+\frac {1}{d-c}\right)\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {M+N}{4}\left (b-a\right) \left (d-c\right), \end{aligned} \] when, \(f\:\) \(I\times J\rightarrow \mathbb {R}\), where \(I\), \(J\) are open intervals in \(\mathbb {R}\), is a mapping such that for \(a,b\in I\), \(c,d\in J\), \(a<b\), \(c<d\), the partial mappings \(f_{y}\left [a,b\right] \rightarrow \mathbb {R}\), \(f_{y}\left (u\right) :=f\left (u,y\right)\) and \(f_{x}\left [c,d\right] \rightarrow \mathbb {R}\), \(f_{x}\left (v\right) :=f\left (x,v\right)\) defined for all \(y\in \left [c,d\right]\) and \(x\in \left [a,b\right]\), are differentiable and \(\left | f_{y}^{^{\prime}}\left (t\right)\right | \leq M\), \(t\in \left [a,b\right]\), \(\left | f_{x}^{^{\prime}}\left (t\right)\right | \leq N\), \(t\in \left [c,d\right]\).
Also, the author’s version of the Cheng theorem [X. L. Cheng, Comput. Math. Appl. 42, No. 1–2, 109–114 (2001; Zbl 0980.26011)] in two variables says in Theorems 2.2 and 2.4 that if \(f\: I\times J\rightarrow \mathbb {R}\), where \(I\), \(J\) are open intervals in \(\mathbb {R}\), is a mapping such that for \(a,b\in I\), \(c,d\in J\), \(a<b\), \(c<d\), the partial mappings \(f_{y}\left [a,b\right] \rightarrow \mathbb {R}\), \(f_{y}\left (u\right) :=f\left (u,y\right)\) and \(f_{x}\left [c,d\right]\rightarrow \mathbb {R}\), \(f_{x}\left (v\right) :=f\left (x,v\right)\) defined for all \(y\in \left [c,d\right]\) and \(x\in \left [a,b\right]\), are differentiable with \(\gamma_{y}\leq f_{y}^{^{\prime}}\left (t\right) \leq \Gamma_{y}\), \(t\in \left [a,b\right]\), \(\gamma_{x}\leq f_{x}^{^{\prime}}\left (t\right) \leq \Gamma_{x}\), \(t\in \left [c,d\right]\) then
\[ \begin{aligned} &\left | \int \limits_{a}^{b}\frac {f\left (x,c\right) +f\left (x,d\right)}{2}{\operatorname {d}}x+\int \limits_{c}^{d}\frac {f\left (a,y\right) +f\left (b,y\right)}{2}{\operatorname {d}}x\right . \\ &\left . -\left (\frac {1}{b-a}+\frac {1}{d-c}\right)\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {\Gamma_{x}+\Gamma_{y}-\gamma_{x}+\gamma_{y}}{8}\left (b-a\right)\left (d-c\right), \end{aligned} \] and \[ \begin{aligned} &\left | \frac {1}{4}\left [\frac {1}{b-a}\int \limits_{a}^{b}\bigl (f\left (x,c\right) +f\left (x,d\right)\bigr) {\operatorname {d}}x+\frac {1}{d-c}\int \limits_{c}^{d}\bigl (f\left (a,y\right) + f\left (b,y\right)\bigr) {\operatorname {d}}y\right]\right . \\ &\left . -\frac {1}{\left (b-a\right)\left (d-c\right)}\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {\left (\left (b-a\right)\left (\Gamma_{y}-\gamma_{y}\right)+\left (d-c\right) \left (\Gamma_{x}-\gamma_{x}\right)\right)}{16}. \end{aligned} \]
Finally in Theorem 2.6, the author proves also that under the assumptions of Theorem 2.1 \[ \begin{aligned} &\left | \frac {1}{2}\left [\int \limits_{a}^{b}f\left (x,\frac {c+d}{2}\right){\operatorname {d}}x+\int \limits_{c}^{d}f\left (\frac {a+b}{2},y\right) {\operatorname {d}}y\right]\right . \\ &\left . -\frac {1}{\left (b-a\right)\left (d-c\right)}\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {M\left (b-a\right) +N\left (d-c\right)}{8} \end{aligned} \] holds.
\[ \begin{aligned} &\left | \int \limits_{a}^{b}\frac {f\left (x,c\right) +f\left (x,d\right)}{2}{\operatorname {d}}x+\int \limits_{c}^{d}\frac {f\left (a,y\right) +f\left (b,y\right)}{2}{\operatorname {d}}x\right . \\ &\left . -\left (\frac {1}{b-a}+\frac {1}{d-c}\right)\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right) \\ \leq {} &\frac {M+N}{2}\left (b-a\right) \left (d-c\right), \end{aligned} \] \[ \begin{aligned} &\left | \frac {1}{4}\left [\frac {1}{b-a}\int \limits_{a}^{b}\bigl (f\left (x,c\right) +f\left (x,d\right)\bigr) {\operatorname {d}}x+\frac {1}{d-c}\int \limits_{c}^{d}\bigl (f\left (a,y\right) + f\left (b,y\right)\bigr) {\operatorname {d}}y\right]\right . \\ &\left . -\frac {1}{\left (b-a\right) \left (d-c\right)}\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {1}{4}\bigl (\left (b-a\right) M+\left (d-c\right) N\bigr), \end{aligned} \] and \[ \begin{aligned} &\left | \int \limits_{a}^{b}f\left (x,\frac {c+d}{2}\right)dx+\int \limits_{c}^{d}f\left (\frac {a+b}{2},y\right) {\operatorname {d}}y\right . \\ &\left . -\left (\frac {1}{b-a}+\frac {1}{d-c}\right)\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {M+N}{4}\left (b-a\right) \left (d-c\right), \end{aligned} \] when, \(f\:\) \(I\times J\rightarrow \mathbb {R}\), where \(I\), \(J\) are open intervals in \(\mathbb {R}\), is a mapping such that for \(a,b\in I\), \(c,d\in J\), \(a<b\), \(c<d\), the partial mappings \(f_{y}\left [a,b\right] \rightarrow \mathbb {R}\), \(f_{y}\left (u\right) :=f\left (u,y\right)\) and \(f_{x}\left [c,d\right] \rightarrow \mathbb {R}\), \(f_{x}\left (v\right) :=f\left (x,v\right)\) defined for all \(y\in \left [c,d\right]\) and \(x\in \left [a,b\right]\), are differentiable and \(\left | f_{y}^{^{\prime}}\left (t\right)\right | \leq M\), \(t\in \left [a,b\right]\), \(\left | f_{x}^{^{\prime}}\left (t\right)\right | \leq N\), \(t\in \left [c,d\right]\).
Also, the author’s version of the Cheng theorem [X. L. Cheng, Comput. Math. Appl. 42, No. 1–2, 109–114 (2001; Zbl 0980.26011)] in two variables says in Theorems 2.2 and 2.4 that if \(f\: I\times J\rightarrow \mathbb {R}\), where \(I\), \(J\) are open intervals in \(\mathbb {R}\), is a mapping such that for \(a,b\in I\), \(c,d\in J\), \(a<b\), \(c<d\), the partial mappings \(f_{y}\left [a,b\right] \rightarrow \mathbb {R}\), \(f_{y}\left (u\right) :=f\left (u,y\right)\) and \(f_{x}\left [c,d\right]\rightarrow \mathbb {R}\), \(f_{x}\left (v\right) :=f\left (x,v\right)\) defined for all \(y\in \left [c,d\right]\) and \(x\in \left [a,b\right]\), are differentiable with \(\gamma_{y}\leq f_{y}^{^{\prime}}\left (t\right) \leq \Gamma_{y}\), \(t\in \left [a,b\right]\), \(\gamma_{x}\leq f_{x}^{^{\prime}}\left (t\right) \leq \Gamma_{x}\), \(t\in \left [c,d\right]\) then
\[ \begin{aligned} &\left | \int \limits_{a}^{b}\frac {f\left (x,c\right) +f\left (x,d\right)}{2}{\operatorname {d}}x+\int \limits_{c}^{d}\frac {f\left (a,y\right) +f\left (b,y\right)}{2}{\operatorname {d}}x\right . \\ &\left . -\left (\frac {1}{b-a}+\frac {1}{d-c}\right)\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {\Gamma_{x}+\Gamma_{y}-\gamma_{x}+\gamma_{y}}{8}\left (b-a\right)\left (d-c\right), \end{aligned} \] and \[ \begin{aligned} &\left | \frac {1}{4}\left [\frac {1}{b-a}\int \limits_{a}^{b}\bigl (f\left (x,c\right) +f\left (x,d\right)\bigr) {\operatorname {d}}x+\frac {1}{d-c}\int \limits_{c}^{d}\bigl (f\left (a,y\right) + f\left (b,y\right)\bigr) {\operatorname {d}}y\right]\right . \\ &\left . -\frac {1}{\left (b-a\right)\left (d-c\right)}\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {\left (\left (b-a\right)\left (\Gamma_{y}-\gamma_{y}\right)+\left (d-c\right) \left (\Gamma_{x}-\gamma_{x}\right)\right)}{16}. \end{aligned} \]
Finally in Theorem 2.6, the author proves also that under the assumptions of Theorem 2.1 \[ \begin{aligned} &\left | \frac {1}{2}\left [\int \limits_{a}^{b}f\left (x,\frac {c+d}{2}\right){\operatorname {d}}x+\int \limits_{c}^{d}f\left (\frac {a+b}{2},y\right) {\operatorname {d}}y\right]\right . \\ &\left . -\frac {1}{\left (b-a\right)\left (d-c\right)}\int \limits_{a}^{b}\!\!\int \limits_{c}^{d}f\left (x,y\right) {\operatorname {d}}x\,{\operatorname {d}}y\right | \\ \leq {} &\frac {M\left (b-a\right) +N\left (d-c\right)}{8} \end{aligned} \] holds.
Reviewer: Shoshana Abramovich (Haifa)
MSC:
| 26A51 | Convexity of real functions in one variable, generalizations |
| 26D15 | Inequalities for sums, series and integrals |
