This work is a generalization of Ostrowski type integral inequalities using a special 4-step quadratic kernel. Some new and useful results are obtained. Applications to Quadrature Rules and special Probability distribution are also evaluated.
Mathematical inequalities constitute a highly versatile field with widespread applications in Mathematics, Statistics, and various other scientific domains. Notably, numerous specialized inequalities, such as the Cauchy-Schwartz inequality, Minkowski’s inequality, Hermite-Hadamard inequality, and the Heisenberg Uncertainty Principle, have significantly influenced the trajectory of science. Owing to their extensive utility, various categories of inequalities have emerged, including integral, differential, and fractional inequalities.
One noteworthy subclass of integral inequalities is the Ostrowski type inequality, which finds broad and remarkable applications across multiple branches of mathematics, encompassing evaluation and analysis (for detailed references, see, for example, [1]-[9]). Ostrowski type inequalities play a pivotal role in diverse mathematical fields, such as error estimation in Numerical Integration and Probability Theory. In the realm of approximation theory, these inequalities serve to gauge the precision of polynomial or spline approximations to a given function within a specified interval.
With this context established, we now proceed to present our primary findings.
We develop a special 4-step quadratic kernel to produce new inequalities of Ostrowski type:
Lemma 1. Let \(f\ \): \(\left[ a,b\right] \rightarrow \mathbb{R}\) be such that \(f^{\text{ }\prime }\) is absolutely continuous on \(\left[ a,b\right]\) for all x\(\in \left[ a,\frac{a+b}{2}\right]\). Then the following identity holds. \[\begin{eqnarray} &&\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t}) \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t} \label{7} \\ &=&2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{ g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \breve{ g}\left( \breve{a}+\flat -x\right) \notag \\ &&+\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{ a}+\flat \right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ &&\breve{g}^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4} \breve{g}^{\text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a}+\flat \right) ^{2}-4\breve{a}\left( \breve{a}-x\right) -4\flat x \right] +\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g} \left( \check{t}\right) d\check{t}. \notag \end{eqnarray} \tag{1}\]
Proof. Define \(P(x,\)ť\()\) as: \[\begin{equation} P(x,\check{t})=\left\{ \begin{array}{ccc} \frac{1}{2}\left( \check{t}-\breve{a}\right) ^{2}, & & \check{t}\in \left( \breve{a},\frac{\breve{a}+x}{2}\right] , \\ & & \\ \frac{1}{2}\left( \check{t}-\frac{3\breve{a}+\flat }{4}\right) ^{2},\text{ } & & \check{t}\in \left( \frac{\breve{a}+x}{2},x\right] , \\ & & \\ \frac{1}{2}\left( \check{t}-\frac{\breve{a}+\flat }{2}\right) ^{2},\text{ } & & \text{\ }\check{t}\in \left( x,\breve{a}+\flat -x\right] , \\ & & \\ \frac{1}{2}\left( \check{t}-\flat \right) ^{2}, & & \check{t}\in \left( \breve{a}+\flat -x,\flat \right] \end{array} \right. \end{equation} \tag{2}\] for all \(x\in \left[ \breve{a},\frac{\breve{a}+\flat }{2}\right] .\)
Consider \[\begin{align*} & \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P\left( x,\check{ t}\right) \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t }=\frac{1}{\flat -\breve{a}}\left[ \int\limits_{\breve{a}}^{\frac{\breve{a}+x }{2}}\frac{1}{2}\left( \check{t}-\breve{a}\right) ^{2}\breve{g}^{\text{ } \prime \prime }\left( \check{t}\right) d\check{t}+\int\limits_{\frac{\breve{a }+x}{2}}^{x}\frac{1}{2}\left( \check{t}-\frac{3\breve{a}+\flat }{4}\right) ^{2}\breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t} \right. \\ & +\left. \int\limits_{x}^{\breve{a}+\flat -x}\frac{1}{2}\left( \check{t}- \frac{\breve{a}+\flat }{2}\right) ^{2}\breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t}+\int\limits_{\breve{a}+\flat -x}^{\flat } \frac{1}{2}\left( \check{t}-\flat \right) ^{2}\breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t}\right] . \end{align*}\] Integrating by parts we get the required result (1) which completes the proof. ◻
Let’s produce new results by applying three different constraints on \(\breve{ g}^{\text{ }\prime\prime}\) and \(\breve{g}^{\text{ }\prime\prime\prime }\).
Theorem 1. Let \(\breve{g}\ \): \(\left[ \breve{a},\flat \right] \rightarrow \mathbb{R}\) is differentiable on \(\left( \breve{a},\flat \right)\), \(\breve{g}^{ \text{ }\prime \text{ }}\)be absolutely continuous on \(\left[ \breve{a},\flat \right]\) and \(\gamma \leq \breve{g}^{\text{ }\prime \prime }(\)ť\()\leq \Gamma\), \(\forall\) ť\(\in \left[ \breve{a},\flat \right] ,\) then \(\forall\) \(x\in \left[ \breve{a},\frac{\breve{a}+\flat }{2}\right] :\) \[\begin{align} & \left\vert 2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \breve{g}\left( \breve{a}+\flat -x\right) \right. \label{34} \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{ a}+\flat \right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ & \breve{g}^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4} \breve{g}^{\text{ }\prime }\left( \breve{a}+\flat -x\right) \notag \\ & \times \left[ \left( \breve{a}+\flat \right) ^{2}-4\breve{a}\left( \breve{a }-x\right) -4\flat x\right] +\frac{\breve{g}^{\text{ }\prime }(\flat )- \breve{g}^{\text{ }\prime }(\breve{a})}{\left( \flat -\breve{a}\right) ^{2}} \notag \times \frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a }\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \notag \\ & \left. -\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g} \left( \check{t}\right) d\check{t}\right\vert \notag \leq \upsilon \left( x\right) \left( \flat -\breve{a}\right) \left( S-\gamma \right) \notag \end{align} \tag{3}\] and \[\begin{align} & \left\vert 2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \breve{g}\left( \breve{a}+\flat -x\right) \right. \label{3} \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{ a}+\flat \right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ & \breve{g}^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4} \breve{g}^{\text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a}+\flat \right) ^{2}-4\breve{a}\left( \breve{a}-x\right) -4\flat x \right] \notag \\ & +\frac{\breve{g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }( \breve{a})}{\left( \flat -\breve{a}\right) ^{2}} \notag \times \frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a }\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \notag \\ & \left. -\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g} \left( \check{t}\right) d\check{t}\right\vert \notag \leq \upsilon \left( x\right) \left( \flat -\breve{a}\right) \left( \Gamma -S\right) , \notag \end{align} \tag{4}\] where \[\begin{equation} S=\frac{\breve{g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }( \breve{a})}{\flat -\breve{a}} \label{4} \end{equation} \tag{5}\] and \[\begin{align} & \upsilon \left( x\right) =\frac{1}{48}\max \left\{ \left\vert 22\breve{a} ^{3}+16x^{3}+63\breve{a}x^{2}-81\breve{a}^{2}x+21\breve{a}^{2}\flat -3\breve{ a}\flat ^{2}+21\flat x^{2}\right. \right. \label{5} \\ & \left. +9\flat ^{2}x-60\breve{a}\flat x\right\vert ,\ \left\vert -26\breve{ a}^{3}-16x^{3}+6\flat ^{3}+45\breve{a}x^{2}-57\breve{a}^{2}x+33\breve{a} ^{2}\flat -33\breve{a}\flat ^{2}+39\flat x^{2}\right. \notag \\ & \left. +24\breve{a}\flat x\right\vert ,\left\vert 25\breve{a} ^{3}+16x^{3}+57\breve{a}x^{2}-81\breve{a}^{2}x+24\breve{a}^{2}\flat -6\breve{ a}\flat ^{2}+27\flat x^{2}+3\flat ^{2}x\right. \notag \\ & \left. -60\breve{a}\flat x\right\vert ,\left\vert -28\breve{a} ^{3}+16x^{3}-69\breve{a}x^{2}-93\breve{a}^{2}x-15\breve{a}^{2}\flat -3\breve{ a}\flat ^{2}-15\flat x^{2}+9\flat ^{2}x\right. \left. \left. +48\breve{a} \flat x\right\vert \right\} . \notag \end{align} \tag{6}\]
Proof. We know that \[\begin{equation*} \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g}^{\text{ } \prime \prime }\left( \check{t}\right) d\check{t}=\frac{\breve{g}^{\text{ } \prime }(\flat )-\breve{g}^{\text{ }\prime }(\breve{a})}{\flat -\breve{a}}. \end{equation*}\] Now consider \[\begin{align*} & \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P\left( x,\check{ t}\right) d\check{t}=\frac{1}{\flat -\breve{a}}\left[ \int\limits_{\breve{a}}^{\frac{\breve{a} +x}{2}}\frac{1}{2}\left( \check{t}-\breve{a}\right) ^{2}d\check{t} +\int\limits_{\frac{\breve{a}+x}{2}}^{x}\frac{1}{2}\left( \check{t}-\frac{3 \breve{a}+\flat }{4}\right) ^{2}d\check{t}\right. \\ & \left. +\int\limits_{x}^{\breve{a}+\flat -x}\frac{1}{2}\left( \check{t}- \frac{\breve{a}+\flat }{2}\right) ^{2}d\check{t}+\int\limits_{\breve{a} +\flat -x}^{\flat }\frac{1}{2}\left( \check{t}-\flat \right) ^{2}d\check{t} \right] \end{align*}\] By integrating , we get \[\begin{align} & \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P\left( x,\check{ t}\right) d\check{t} \label{6} \\ & =\frac{1}{6\left( \flat -\breve{a}\right) }\left[ \left( \frac{\breve{a}+x }{2}-\breve{a}\right) ^{3}+\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{3}+\left( \frac{x+2\breve{a}+\flat }{2}\right) ^{3}\right. \notag \\ & \left\{ \left( \breve{a}+\flat -x-\frac{\breve{a}+\flat }{2}\right) ^{3}-\left( x-\frac{\breve{a}+\flat }{2}\right) ^{3}\right\} -\left( \breve{a }-x\right) ^{3} \notag \\ & =\frac{1}{6\left( \flat -\breve{a}\right) }\left[ \left( \frac{x-\breve{a} }{2}\right) ^{3}+\left( \frac{2x-3\breve{a}-\flat }{2}\right) ^{3}-\left( \frac{x-2\breve{a}-\flat }{2}\right) ^{3}\right. \notag \\ & \left. +\left( \frac{\breve{a}+\flat -2x}{2}\right) ^{3}-\left( \frac{2x- \breve{a}-\flat }{2}\right) ^{3}-\left( \breve{a}-x\right) ^{3}\right] \notag \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left[ \left( x-\breve{a} \right) ^{3}+\left( 2x-3\breve{a}-\flat \right) ^{3}-\left( x-2\breve{a} -\flat \right) ^{3}\right. \notag \\ & \left. -\left( 2x-\breve{a}-\flat \right) ^{3}-8\left( \breve{a}-x\right) ^{3}\right] \notag \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a} \right) ^{3}+8\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] . \notag \end{align} \tag{7}\] Now we proceed by considering \[\begin{align*} & \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t}) \breve{g}^{\text{ }\prime \prime }(\check{t})d\check{t}-\frac{1}{\left( \flat -\breve{a}\right) ^{2}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t})d \check{t}\int\limits_{\breve{a}}^{\flat }\breve{g}^{\text{ }\prime \prime }( \check{t})d\check{t} \\ & =2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{ g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \breve{ g}\left( \breve{a}+\flat -x\right) \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{ a}+\flat \right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \end{align*}\] \[\begin{align} & \breve{g}^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4} \breve{g}^{\text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a}+\flat \right) ^{2}-4\breve{a}\left( \breve{a}-x\right) -4\flat x \right] \notag \\ & +\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g}\left( \check{t}\right) -\frac{\breve{g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }(\breve{a})}{\flat -\breve{a}} \notag \\ & \times \frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a }\right) ^{3}+8\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] . \notag \end{align}\] Define \[\begin{align*} & R_{n}\left( x\right) =\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t}) \breve{g}^{\text{ }\prime \prime }(\check{t})d\check{t}-\frac{1}{\left( \flat -\breve{a}\right) ^{2}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t})d \check{t}\int\limits_{\breve{a}}^{\flat }\breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t}. \end{align*}\] Let \(C\in R\) be a constant of arbitrary nature, then we have \[\begin{align} & R_{n}\left( x\right) \label{20} \\ & =\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\left( \breve{g} ^{\text{ }\prime \prime }(\check{t})-C\right) \left[ P(x,\check{t})-\frac{1}{ \flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,s)ds\right] d\check{t} \notag \end{align} \tag{8}\] which leads to the following relation: \[\begin{align} & \left\vert R_{n}\left( x\right) \right\vert \label{21} \\ & \leq \frac{1}{\flat -\breve{a}}\underset{\check{t}\in \left[ \breve{a} ,\flat \right] }{\max }\left\vert P(x,\check{t})-\frac{1}{\flat -\breve{a}} \int\limits_{\breve{a}}^{\flat }P(x,s)ds\right\vert \notag \\ & \times \int\limits_{\breve{a}}^{\flat }\left\vert \breve{g}^{\text{ } \prime \prime }(\check{t})-C\right\vert d\check{t}. \notag \end{align} \tag{9}\] We also have \[\begin{align*} & \max \left\vert P(x,\check{t})-\frac{1}{\flat -\breve{a}}\int\limits_{ \breve{a}}^{\flat }P(x,s)ds\right\vert \\ & =\max \left\{ \left\vert \frac{1}{2}\left( \frac{x-\breve{a}}{2}\right) ^{2}-\frac{\lambda (x)}{\flat -\breve{a}}\right\vert ,\left\vert \frac{1}{2} \left( x-\frac{3\breve{a}+\flat }{2}\right) ^{2}-\frac{\lambda (x)}{\flat – \breve{a}}\right\vert ,\right. \\ & \left. \left\vert \frac{1}{2}\left( x-\frac{\breve{a}-\flat }{2}\right) ^{2}-\frac{\lambda \left( x\right) }{\flat -\breve{a}}\right\vert ,\left\vert \frac{\lambda (x)}{\flat -\breve{a}}\right\vert \right\} \end{align*}\] where \[\begin{align*} & \lambda \left( x\right) \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a} \right) ^{3}+8\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] . \end{align*}\] Consider \[\begin{align} & \left\vert \frac{1}{2}\left( \frac{x-\breve{a}}{2}\right) ^{2}-\frac{ \lambda (x)}{\flat -\breve{a}}\right\vert \label{8} \\ & =\left\vert \frac{x^{2}+\breve{a}^{2}-2\breve{a}x}{8}-\frac{1}{48\left( \flat -\breve{a}\right) }\right. \notag \\ & \left. \left. \left[ 9\left( x-\breve{a}\right) ^{3}\right. +8\left( x- \frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \right\vert \notag \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left\vert 22\breve{a} ^{3}+16x^{3}+63\breve{a}x^{2}-81\breve{a}^{2}x+21\breve{a}^{2}\flat -3\breve{ a}\flat ^{2}\right. \notag \\ & \left. +21\flat x^{2}+9\flat ^{2}x-60\breve{a}\flat x\right\vert , \notag \end{align} \tag{10}\] similarly, we obtain \[\begin{align} & \left\vert \frac{1}{2}\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{2}- \frac{\lambda (x)}{\flat -\breve{a}}\right\vert \notag \\ & =\left\vert \frac{4x^{2}+\breve{a}^{2}+\flat ^{2}-4\breve{a}x+4\flat x-2 \breve{a}\flat }{8}-\frac{1}{48\left( \flat -\breve{a}\right) }\right. \\ & \left. \left. \left[ 9\left( x-\breve{a}\right) ^{3}\right. +8\left( x- \frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \right\vert \notag \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left\vert -26\breve{a} ^{3}-16x^{3}+6\flat ^{3}+45\breve{a}x^{2}-57\breve{a}^{2}x+33\breve{a} ^{2}\flat \right. \notag \\ & \left. -33\breve{a}\flat ^{2}+39\flat x^{2}+24\breve{a}\flat x\right\vert \notag \end{align} \tag{11}\] and \[\begin{align} & \left\vert \frac{1}{2}\left( x-\frac{\breve{a}-\flat }{2}\right) ^{2}- \frac{\lambda \left( x\right) }{\flat -\breve{a}}\right\vert \label{10} \\ & =\left\vert \frac{4x^{2}+\breve{a}^{2}+\flat ^{2}-4\breve{a}x+4\flat x-2 \breve{a}\flat }{8}-\frac{1}{48\left( \flat -\breve{a}\right) }\right. \notag \\ & \left. \left. \left[ 9\left( x-\breve{a}\right) ^{3}\right. +8\left( x- \frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \right\vert \notag \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left\vert 25\breve{a} ^{3}+16x^{3}+57\breve{a}x^{2}-81\breve{a}^{2}x+24\breve{a}^{2}\flat -6\breve{ a}\flat ^{2}\right. \notag \\ & \left. +27\flat x^{2}+3\flat ^{2}x-60\breve{a}\flat x\right\vert . \notag \end{align} \tag{12}\] We can also conclude: \[\begin{align} & \left\vert \frac{\lambda (x)}{\flat -\breve{a}}\right\vert \label{11} \\ & =\frac{1}{48\left( \flat -\breve{a}\right) }\left\vert -28\breve{a} ^{3}+16x^{3}-69\breve{a}x^{2}-93\breve{a}^{2}x-15\breve{a}^{2}\flat \right. \notag \\ & \left. -3\breve{a}\flat ^{2}-15\flat x^{2}+9\flat ^{2}x+48\breve{a}\flat x\right\vert . \notag \end{align} \tag{13}\] With the combination of (10) till (13) we get (6).
Furthermore, \[\begin{equation} \int\limits_{\breve{a}}^{\flat }\left\vert \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\gamma \right\vert d\check{t}=\left( S-\gamma \right) \left( \flat -\breve{a}\right) \label{12} \end{equation} \tag{14}\] \[\begin{equation} \int\limits_{\breve{a}}^{\flat }\left\vert \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\Gamma \right\vert d\check{t}=\left( \Gamma -S\right) \left( \flat -\breve{a}\right) . \label{13} \end{equation} \tag{15}\] We have obtained (3) and (4) by combining(7) till (15). ◻
Theorem 2. Let \(\breve{g}\ \): \(\left[ \breve{a},\flat \right] \rightarrow \mathbb{R}\) be a three times differentiable function on \(\left( \breve{a},\flat \right) .\) If \(\breve{g}^{\text{ }\prime \prime \prime }\in L^{2}\left[ \breve{a},\flat \right] ,\) then\(\forall\) \(x\in \left[ \breve{a},\frac{ \breve{a}+\flat }{2}\right] :\) \[\begin{align} & \left\vert 2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \breve{g}\left( \breve{a}+\flat -x\right) \right. \label{31} \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{ a}+\flat \right) ^{2}-\right. \notag \\ & \left. \left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ & \times \breve{g}^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) + \frac{1}{4}\breve{g}^{\text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a}+\flat \right) ^{2}-4\breve{a}\left( \breve{a}-x\right) \right. \notag \\ & 4\flat x+\frac{\breve{g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ } \prime }(\breve{a})}{\left( \flat -\breve{a}\right) ^{2}} \notag \\ & \times \frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a }\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}\right. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \left. -\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g} \left( \check{t}\right) d\check{t}\right\vert \notag \end{align} \tag{16}\] \[\begin{align} & \leq \frac{1}{\pi }\left\Vert \breve{g}^{\text{ }\prime \prime \prime }\right\Vert _{2}\left\{ \frac{33}{160}\left( x-\breve{a}\right) ^{5}+32\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{5}\right. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) ^{5}+64\left( x- \frac{\breve{a}+\flat }{2}\right) ^{5}\right\} \times \frac{1}{\flat -\breve{ a}}\left\{ 9\left( x-\breve{a}\right) ^{3}\right. \notag \\ & \left. +\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) ^{3}\right\} \notag \\ & \left. -\frac{1}{\flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) ^{3}\right\} \right] ^{\frac{1}{2}}. \notag \end{align}\]
Proof. Let \(R_{n}\left( x\right)\) be given as \[\begin{align} & R_{n}\left( x\right) \label{15} =\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t}) \breve{g}^{\text{ }\prime \prime }(\check{t})d\check{t}-\frac{1}{\left( \flat -\breve{a}\right) ^{2}}\int\limits_{\breve{a}}^{\flat }P(x,\check{t})d \check{t}\int\limits_{\breve{a}}^{\flat }\breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) d\check{t} \notag \\ & =\left[ 2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \right. \notag \\ & \breve{g}\left( \breve{a}+\flat -x\right) +\frac{1}{4}\left( 8\breve{a} ^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{a}+\flat \right) ^{2}\right. \notag \\ & \left. -\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \breve{g} ^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4}\breve{g}^{ \text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a} +\flat \right) ^{2}\right. \notag \\ & \left. -4\breve{a}\left( \breve{a}-x\right) -4\flat x\right] +\frac{\breve{ g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }(\breve{a})}{\left( \flat -\breve{a}\right) ^{2}} \notag \\ & \times \frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x-\breve{a }\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}\right. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] – \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g}\left( \check{t}\right) d\check{t}. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] – \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g}\left( \check{t}\right) d\check{t}. \notag \end{align}\] By choosing \(C=\breve{g}^{\text{ }\prime \prime }\left( \frac{\breve{a} +\flat }{2}\right)\) and using Cauchy’s Inequality, we get \[\begin{align*} & \left\vert R_{n}\left( x\right) \right\vert \\ & \leq \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\left\vert \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\breve{g}^{\text{ }\prime \prime }\left( \frac{\breve{a}+\flat }{2}\right) \right\vert \left\vert p\left( x,\check{t}\right) \right. \\ & \left. -\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }p(x,s)ds\right\vert d\check{t} \\ & \leq \frac{1}{\flat -\breve{a}}\left[ \int\limits_{\breve{a}}^{\flat }\left( \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\breve{g} ^{\text{ }\prime \prime }\left( \frac{\breve{a}+\flat }{2}\right) \right) ^{2}d\check{t}\right] ^{\frac{1}{2}} \\ & \times \left[ \int\limits_{\breve{a}}^{\flat }\left( P(x,\check{t})-\frac{1 }{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,s)ds\right) ^{2}d \check{t}\right] ^{\frac{1}{2}}. \end{align*}\] In the same way, using Diaz-Metcālf inequality, we obtain \[\begin{equation} \int\limits_{\breve{a}}^{\flat }\left( \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\breve{g}^{\text{ }\prime \prime }\left( \frac{ \breve{a}+\flat }{2}\right) \right) ^{2}d\check{t}\leq \frac{\left( \flat – \breve{a}\right) ^{2}}{\pi ^{2}}\left\Vert \breve{g}^{\text{ }\prime \prime \prime }\right\Vert _{2}^{2} \label{17} \end{equation} \tag{17}\] and \[\begin{align} & \int\limits_{\breve{a}}^{\flat }\left( p(x,\check{t})-\frac{1}{\flat – \breve{a}}\int\limits_{\breve{a}}^{\flat }p(x,s)ds\right) ^{2}d\check{t} \label{18} \\ =\int\limits_{\breve{a}}^{\flat }P(x,\check{t})^{2}d\check{t}& -\frac{1}{ \flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3 \breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right\} \notag \\ & =\frac{1}{160}\left[ \left\{ 33\left( x-\breve{a}\right) ^{5}+32\left( x- \frac{3\breve{a}+\flat }{2}\right) ^{5}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{5}\right. \right. \notag \\ & \left. -64\left( x-\frac{\breve{a}+\flat }{2}\right) \right\} ^{5} \notag \\ & -\frac{1}{\flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right\} \notag \end{align} \tag{18}\] Therefore, using the above relations ([15])-(18), we obtain (16) which completes proof. ◻
Theorem 3. Let \(\breve{g}:[\)ă\(;\) \(\flat ]\rightarrow R\) be a function which is absolutely continuous on \((\)ă\(,\flat )\) such that \(\breve{g}^{\text{ } \prime \prime }\in L^{2}[\)ă\(,\flat ]\). Then \(\forall\) \(x\in \left[ \breve{a},\frac{\breve{a}+\flat }{2}\right] :\) \[\begin{align} & \left\vert 2\breve{a}\breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \breve{g}\left( \breve{a}+\flat -x\right) \right. \notag \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{ a}+\flat \right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ & \breve{g}^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4} \breve{g}^{\text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a}+\flat \right) ^{2}-4\breve{a}\left( \breve{a}-x\right) -4\flat x \right] \notag \\ & +\frac{\breve{g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }( \breve{a})}{\left( \flat -\breve{a}\right) ^{2}}\times \frac{1}{48\left( \flat -\breve{a}\right) } \notag \\ \times & \left. \left[ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3 \breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] -\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a} }^{\flat }\breve{g}\left( \check{t}\right) d\check{t}\right\vert \notag \\ & \leq \frac{\sqrt{\sigma \left( \breve{g}^{\text{ }\prime \prime }\right) } }{\flat -\breve{a}} \notag \\ & \times \left[ \left\{ \frac{33}{160}\left( x-\breve{a}\right) ^{5}+32\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{5}-\left( x-\left( 2 \breve{a}+\flat \right) \right) ^{5}\right. \right. \\ & \left. 64\left( x-\frac{\breve{a}+\flat }{2}\right) ^{5}\right\} \times \frac{1}{\flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x- \frac{3\breve{a}+\flat }{2}\right) ^{3}\right. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) \right\} ^{3} \notag \\ & \left. -\frac{1}{\flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) ^{3}\right\} \right] ^{\frac{1}{2}} \notag \end{align} \tag{19}\] where \[\begin{align*} \sigma \left( \breve{g}^{\text{ }\prime \prime }\right) & =\left\Vert \breve{ g}^{\text{ }\prime \prime }\right\Vert _{2}^{2}-\frac{\left( \breve{g}^{ \text{ }\prime }\left( \flat \right) -\breve{g}^{\text{ }\prime }\left( \breve{a}\right) \right) ^{2}}{\flat -\breve{a}} \\ & =\left\Vert \breve{g}^{\text{ }\prime \prime }\right\Vert _{2}^{2}-S^{2}\left( \flat -\breve{a}\right) , \end{align*}\] where \[\begin{equation*} S=\frac{\breve{g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }( \breve{a})}{\flat -\breve{a}}. \end{equation*}\]
Proof. Let \(R_{n}\left( x\right)\) be given as in ([15]), By choosing \(C= \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\breve{g}^{\text{ } \prime \prime }\left( s\right) ds\) in (8) and using Cauchy’s inequality and (18), we get \[\begin{align*} & \left\vert R_{n}\left( x\right) \right\vert \\ & \leq \frac{1}{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }\left\vert \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\breve{g}^{\text{ }\prime \prime }\left( \frac{\breve{a}+\flat }{2}\right) \right\vert \left\vert P(x,\check{t})-\frac{1}{\flat -\breve{a}}\int\limits_{\breve{a} }^{\flat }P(x,s)ds\right\vert d\check{t} \\ & \leq \frac{1}{\flat -\breve{a}}\left[ \int\limits_{\breve{a}}^{\flat }\left( \breve{g}^{\text{ }\prime \prime }\left( \check{t}\right) -\breve{g} ^{\text{ }\prime \prime }\left( \frac{\breve{a}+\flat }{2}\right) \right) ^{2}d\check{t}\right] ^{\frac{1}{2}} \\ & \times \left[ \int\limits_{\breve{a}}^{\flat }\left( P(x,\check{t})-\frac{1 }{\flat -\breve{a}}\int\limits_{\breve{a}}^{\flat }P(x,s)ds\right) ^{2}d \check{t}\right] ^{\frac{1}{2}}. \end{align*}\] \[\begin{align} & =\frac{\sqrt{\sigma \left( \breve{g}^{\text{ }\prime \prime }\right) }}{ \flat -\breve{a}}\left[ \left\{ \frac{33}{160}\left( x-\breve{a}\right) ^{5}+32\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{5}\right. \right. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) ^{5}64\left( x- \frac{\breve{a}+\flat }{2}\right) ^{5}\right\} \times \frac{1}{\flat -\breve{ a}}\left\{ 9\left( x-\breve{a}\right) ^{3}\right. \notag \\ & \left. +\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) \right\} ^{3} \notag \\ & \left. -\frac{1}{\flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) ^{3}\right\} \right] ^{\frac{1}{2}}. \notag \end{align}\] which completes the proof. ◻
Let \(I_{n}:\) \(=\) ă\(=x_{0}<x_{1}<x_{2}<….<x_{n-1}<x_{n}\) \(\ =\flat\) be a distribution of the interval \(\left[ \breve{a},\flat\right]\) and \(\mathbf{ \xi}=\left( \xi_{0},\xi_{1},….\xi_{n-1}\right)\) a sequence of intermediate points \(\xi_{i}\in\left[ x_{i},x_{i+1}\right]\) and \(h=x_{i+1}-x_{i\text{ \ }}\) \(\ \left( i\text{ }=0,1,…..n-1\right)\).
Then: \[\begin{align} & \breve{a}_{n}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right) \label{24} \\ & =\left\vert 2x_{i}\breve{g}\left( \xi _{i}\right) -\left( x_{i}+x_{i+1}\right) \breve{g}\left( \fra c{x_{i}+\xi {i}}{2}\right) +h{i} \breve{g}\left( x_{i}+x_{i+1}-\xi _{i}\right) \right. \notag \\ & +\frac{1}{4}\left( 8x_{i}^{2}-8x_{i}\xi {i}+4x{i}x_{i+1}\right) \breve{g} ^{\text{ }\prime }\left( \xi _{i}\right) -\frac{1}{4}\left[ \left( 2x_{i}+x_{i+1}\right) ^{2}\right. \notag \\ & \left. -\left( x_{i}^{2}+2x_{i}\xi {i}+2x{i+1}\xi _{i}\right) \right] \notag \\ & \breve{g}^{\text{ }\prime }\left( \frac{x_{i}+\xi _{i}}{2}\right) +\frac{1 }{4}\breve{g}^{\text{ }\prime }\left( x_{i}+x_{i+1}-\xi _{i}\right) \left[ \left( x_{i}+x_{i+1}\right) ^{2}-4x_{i}\left( x_{i}-\xi _{i}\right) \right. \notag \\ & \left. -4x_{i+1}\xi _{i}\right] +\frac{\breve{g}^{\text{ }\prime }(x_{i+1})-\breve{g}^{\text{ }\prime }(x_{i})}{h_{i}^{2}} \notag \\ & \times \frac{1}{48h_{i}}\left[ 9\left( \xi {i}-x{i}\right) ^{3}+\left( 8\xi {i}-\frac{3x{i}+x_{i+1}}{2}\right) ^{3}\right. \notag \\ & \left. -\left( \xi {i}-\left( 2x{i}+x_{i+1}\right) \right) ^{3}\right] \notag \end{align} \tag{20}\] for all \(\xi {i}\in \left[ x{i},\frac{x_{i}+x_{i+1}}{2}\right] .\)
Theorem 4. Let \(\breve{g}:[\)ă\(;\flat ]\rightarrow R\) be such that \(\breve{g}^{\text{ }\prime }\) is absolutely continuous function. If \(\breve{g}^{\text{ }\prime \prime }\in L^{1}\left[ \breve{a},\flat \right]\) and \(\gamma \leq \breve{g} ^{\text{ }\prime \prime }(x)\leq \Gamma\) , then : \[\begin{equation*} \int\limits_{\breve{a}}^{\flat }\breve{g}(\check{t})d\check{t}=\breve{a} {n}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi },I{n}\right) +R_{n}^{1}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right) , \end{equation*}\] \[\begin{equation*} \int\limits_{\breve{a}}^{\flat }\breve{g}(\check{t})d\check{t}=\breve{a} {n}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi },I{n}\right) +R_{n}^{2}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right) . \end{equation*}\] Where ă\(_{n}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right)\) is given in (20) and remainder satisfies: \[\begin{equation} \left\vert R_{n}^{1}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi },I_{n}\right) \right\vert \leq \left( S-\gamma \right) \underset{i=0}{ \overset{n-1}{\sum }}h_{i}^{2}\upsilon \left( \xi _{i}\right) \label{22} \end{equation} \tag{21}\] and \[\begin{equation} \left\vert R_{n}^{2}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi },I_{n}\right) \right\vert \leq \left( \Gamma -S\right) \underset{i=0}{ \overset{n-1}{\sum }}h_{i}^{2}\upsilon \left( \xi _{i}\right) . \label{23} \end{equation} \tag{22}\]
Theorem 5. Let \(\breve{g}:[\)ă\(;\flat ]\rightarrow R\)be a mapping which is three times differentiable on \((\)ă\(,\flat )\) with \(\breve{g}^{\text{ }\prime \prime \prime }\) \(\in\) \(L^{2}[\)ă\(,\flat ]\). Then: \[\begin{equation*} \int\limits_{\breve{a}}^{\flat }\breve{g}(\check{t})d\check{t}=\breve{a} {n}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi },I{n}\right) +R_{n}^{3}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right) \end{equation*}\] where ă\(_{n}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right)\) is defined by formula (20) and the remainder \(R_{n}^{3}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{\xi } ,I_{n}\right)\) satisfies the estimation \[\begin{align} & \left\vert R_{n}^{3}\left( \breve{g},\breve{g}^{\text{ }\prime },\mathbf{ \xi },I_{n}\right) \right\vert \label{25} \\ & \leq \frac{1}{\pi }\left\Vert \breve{g}^{\text{ }\prime \prime \prime }\right\Vert _{2}\left\{ \left[ \frac{33}{160}\left( \mathbf{\xi } {i}-x{i}\right) ^{5}+32\left( \mathbf{\xi }{i}-\frac{3x{i}+x_{i+1}}{2} \right) ^{5}\right. \right. \notag \\ & -\left( \mathbf{\xi }{i}-\left( 2x{i}+x_{i+1}\right) \right) ^{5}\left. -64\left( \mathbf{\xi }{i}-\frac{x{i}+x_{i+1}}{2}\right) ^{5}\right\} \notag \\ & \times \frac{1}{h_{i}}\left\{ 9\left( \mathbf{\xi }{i}-x{i}\right) ^{3}+\left( 8\mathbf{\xi }{i}-\frac{3x{i}+x_{i+1}}{2}\right) ^{3}\right. \left. -\left( \mathbf{\xi }{i}-\left( 2x{i}+x_{i+1}\right) \right) ^{3}\right\} \notag \\ & \left[ -\frac{1}{h_{i}}9\left( \mathbf{\xi }{i}-x{i}\right) ^{3}+\left( 8 \mathbf{\xi }{i}-\frac{3x{i}+x_{i+1}}{2}\right) ^{3}\right. \notag \\ & -\left. \left( \mathbf{\xi }{i}-\left( 2x{i}+x_{i+1}\right) \right) \right] ^{\frac{1}{2}}. \notag \end{align} \tag{23}\]
As we know that ” If \(X\) is a random variable which has values in the interval \(\left[ \breve{a},\flat\right]\) with the probability density function \(\breve{g}\ \): \(\left[ \breve{a},\flat\right] \rightarrow\left[ 0,1 \right]\) and cumulative distribution function \[\begin{equation} \breve{g}\left( x\right) =\Pr\left( X\leq x\right) =\int_{\breve{a}}^{x} \breve{g}\left( \check{t}\right) d\check{t}, \label{27} \end{equation} \tag{24}\] \[\begin{equation} \breve{g}\left( \flat\right) =\Pr\left( X\leq\flat\right) =\overset{\flat }{ \underset{\breve{a}}{\int}}\breve{g}\left( u\right) du=1". \label{28} \end{equation} \tag{25}\]
Theorem 6. Using the conditions of Theorem 2, we obtain following inequality which holds \[\begin{align} & \left\vert \frac{\flat-E\left( X\right) }{\flat-\breve{a}}-2\breve {a} \breve{g}\left( x\right) -\left( \breve{a}+\flat\right) \breve {g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat\right) \breve{g}\left( \breve{a}+\flat-x\right) \right. \label{29} \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat\right) \breve{g}^{\text{ }\prime}\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{a }+\flat\right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ & \breve{g}^{\text{ }\prime}\left( \frac{\breve{a}+x}{2}\right) +\frac {1}{4} \breve{g}^{\text{ }\prime}\left( \breve{a}+\flat-x\right) \notag \\ & \times\left[ \left( \breve{a}+\flat\right) ^{2}-4\breve{a}\left( \breve{a} -x\right) -4\flat x\right] +\frac{\breve{g}^{\text{ }\prime}(\flat)-\breve{g} ^{\text{ }\prime}(\breve{a})}{\left( \flat-\breve{a}\right) ^{2}} \notag \\ & \times\left. \frac{1}{48\left( \flat-\breve{a}\right) }\left[ 9\left( x- \breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat}{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat\right) \right) ^{3}\right] \right\vert \notag \\ & \leq\upsilon\left( x\right) \left( \flat-\breve{a}\right) \left( S-\gamma\right) \notag \end{align} \tag{26}\] \[\begin{align} & \left\vert \frac{\flat-E\left( X\right) }{\flat-\breve{a}}-2\breve {a} \breve{g}\left( x\right) -\left( \breve{a}+\flat\right) \breve {g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat\right) \breve{g}\left( \breve{a}+\flat-x\right) \right. \label{30} \\ & +\frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat\right) \breve{g}^{\text{ }\prime}\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{a }+\flat\right) ^{2}-\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \notag \\ & \breve{g}^{\text{ }\prime}\left( \frac{\breve{a}+x}{2}\right) +\frac {1}{4} \breve{g}^{\text{ }\prime}\left( \breve{a}+\flat-x\right) \notag \\ & \times\left[ \left( \breve{a}+\flat\right) ^{2}-4\breve{a}\left( \breve{a} -x\right) -4\flat x\right] +\frac{\breve{g}^{\text{ }\prime}(\flat)-\breve{g} ^{\text{ }\prime}(\breve{a})}{\left( \flat-\breve{a}\right) ^{2}} \notag \\ & \times\left. \frac{1}{48\left( \flat-\breve{a}\right) }\left[ 9\left( x- \breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat}{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat\right) \right) ^{3}\right] \right\vert \notag \\ & \leq\upsilon\left( x\right) \left( \flat-\breve{a}\right) \left( \Gamma-S\right) \notag \end{align} \tag{27}\] for all \(x\). Where \(E\left( X\right)\) is the expectation of \(X\).
Theorem 7. With the conditions of Theorem 5, we have the following inequality which holds \[\begin{align*} & \left\vert \frac{\flat-E\left( X\right) }{\flat-\breve{a}}-2\breve {a} \breve{g}\left( x\right) -\left( \breve{a}+\flat\right) \breve {g}\left( \frac{\breve{a}+x}{2}\right) \right. \\ & +\left( \breve{a}-\flat\right) \breve{g}\left( \breve{a}+\flat-x\right) + \frac{1}{4}\left( 8\breve{a}^{2}-8\breve{a}x+4\breve{a}\flat\right) \breve{g} ^{\text{ }\prime}\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{a} +\flat\right) ^{2}\right. \\ & \left. -\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \breve{g} ^{\text{ }\prime}\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4}\breve{g}^{ \text{ }\prime}\left( \breve{a}+\flat-x\right) \left[ \left( \breve{a} +\flat\right) ^{2}\right. \end{align*}\] \[\begin{align*} & -4\breve{a}\left( \breve{a}-x\right) -4\flat x+\frac{\breve{g}^{\text{ } \prime}(\flat)-\breve{g}^{\text{ }\prime}(\breve{a})}{\left( \flat-\breve {a} \right) ^{2}} \\ & \left. \times\frac{1}{48\left( \flat-\breve{a}\right) }\left[ 9\left( x- \breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat}{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat\right) \right) ^{3}\right] \right\vert \\ & \leq\frac{1}{\pi}\left\Vert \breve{g}^{\text{ }\prime\prime\prime }\right\Vert _{2}\left[ \left\{ \frac{33}{160}\left( x-\breve{a}\right) ^{5}+32\left( x-\frac{3\breve{a}+\flat}{2}\right) ^{5}-\left( x-\left( 2 \breve{a}+\flat\right) \right) ^{5}\right. \right. \\ & \left. 64\left( x-\frac{\breve{a}+\flat}{2}\right) ^{5}\right\} \times \frac{1}{\flat-\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x- \frac{3\breve{a}+\flat}{2}\right) ^{3}\right. \\ & \left. -\left( x-\left( 2\breve{a}+\flat\right) \right) ^{3}\right\} \\ & \left. -\frac{1}{\flat-\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat}{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat\right) \right) ^{3}\right\} \right] ^{\frac{1}{2}}. \end{align*}\] for all \(x\in\left[ \breve{a},\frac{\breve{a}+\flat}{2}\right]\), where \(E\left( X\right)\) is the expectation of \(X\).
Theorem 8. By using same conditions as of Theorem 9, we obtain the inequality: \[\begin{align} & \left\vert \frac{\flat -E\left( X\right) }{\flat -\breve{a}}-2\breve{a} \breve{g}\left( x\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{\breve{a}+x}{2}\right) +\left( \breve{a}-\flat \right) \right. \label{35} \\ & \breve{g}\left( \breve{a}+\flat -x\right) +\frac{1}{4}\left( 8\breve{a} ^{2}-8\breve{a}x+4\breve{a}\flat \right) \breve{g}^{\text{ }\prime }\left( x\right) -\frac{1}{4}\left[ \left( 2\breve{a}+\flat \right) ^{2}\right. \notag \\ & \left. -\left( \breve{a}^{2}+2\breve{a}x+2\flat x\right) \right] \breve{g} ^{\text{ }\prime }\left( \frac{\breve{a}+x}{2}\right) +\frac{1}{4}\breve{g}^{ \text{ }\prime }\left( \breve{a}+\flat -x\right) \left[ \left( \breve{a} +\flat \right) ^{2}\right. \notag \\ & \left. -4\breve{a}\left( \breve{a}-x\right) -4\flat x\right] +\frac{\breve{ g}^{\text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }(\breve{a})}{\left( \flat -\breve{a}\right) ^{2}} \notag \\ & \left. \times \frac{1}{48\left( \flat -\breve{a}\right) }\left[ 9\left( x- \breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2\breve{a}+\flat \right) \right) ^{3}\right] \right\vert \notag \end{align} \tag{28}\] \[\begin{align} & \leq \frac{\sqrt{\sigma \left( \breve{g}^{\text{ }\prime \prime }\right) } }{\flat -\breve{a}}\left[ \left\{ \frac{33}{160}\left( x-\breve{a}\right) ^{5}+32\left( x-\frac{3\breve{a}+\flat }{2}\right) ^{5}\right. \right. \notag \\ & \left. -\left( x-\left( 2\breve{a}+\flat \right) \right) ^{5}64\left( x- \frac{\breve{a}+\flat }{2}\right) ^{5}\right\} \times \frac{1}{\flat -\breve{ a}}\left\{ 9\left( x-\breve{a}\right) ^{3}\right. \notag \\ & \left. +\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) \right\} ^{3} \notag \\ & \left. -\frac{1}{\flat -\breve{a}}\left\{ 9\left( x-\breve{a}\right) ^{3}+\left( 8x-\frac{3\breve{a}+\flat }{2}\right) ^{3}-\left( x-\left( 2 \breve{a}+\flat \right) \right) ^{3}\right\} \right] ^{\frac{1}{2}}. \notag \end{align}\]
Proof. Keeping in mind, the conditions of Theorem 19, if we put \(x=\frac{3\breve{a} +\flat }{4}\) in (16), then: \[\begin{align*} & \left\vert \frac{\flat -E\left( X\right) }{\flat -\breve{a}}-2\breve{a} \breve{g}\left( \frac{3\breve{a}+\flat }{4}\right) -\left( \breve{a}+\flat \right) \breve{g}\left( \frac{7\breve{a}+\flat }{4}\right) +\left( \breve{a} -\flat \right) \right. \\ & \breve{g}\left( \frac{\breve{a}+3\flat }{4}\right) +\frac{1}{2}\breve{a} \left( \breve{a}+\flat \right) \breve{g}^{\text{ }\prime }\left( \frac{3 \breve{a}+\flat }{4}\right) -\frac{1}{8}\left[ \left( 2\breve{a}+\flat \right) ^{2}-\breve{a}^{2}\right] \\ & \left. \breve{g}^{\text{ }\prime }\left( \frac{7\breve{a}+\flat }{8} \right) -\frac{1}{2}\flat \left( \flat -\breve{a}\right) +\frac{\breve{g}^{ \text{ }\prime }(\flat )-\breve{g}^{\text{ }\prime }(\breve{a})}{\left( \flat -\breve{a}\right) ^{2}}\right\vert \\ & \times \frac{1}{3072\left( \flat -\breve{a}\right) }\left[ 9\left( \flat – \breve{a}\right) ^{3}+216\left( 3\breve{a}+\flat \right) ^{3}-\left( 5\breve{ a}+3\flat \right) ^{3}\right] \\ & \leq \frac{\sqrt{\sigma \left( \breve{g}^{\text{ }\prime \prime }\right) } }{\flat -\breve{a}}\left[ \frac{33}{160}\left( \frac{\flat -\breve{a}}{4} \right) ^{5}-\left( 3\breve{a}+\flat \right) ^{5}-\left( 5\breve{a}+3\flat \right) ^{5}\right] \\ & \times \frac{1}{\flat -\breve{a}}\left[ 9\left( \flat -\breve{a}\right) ^{3}+216\left( 3\breve{a}+\flat \right) ^{3}-\left( 5\breve{a}+3\flat \right) ^{3}\right] ^{\frac{1}{2}}. \end{align*}\] Hence the required inequality . ◻
This work explored the Ostrwoski-type Inequalities by using the innovative 4-step Quadratic kernel has yielded profound insights into the realm of Mathematics, precisely to Statistics and Numerical Analysis. Through a tedious journey of theoretical development and analysis, this work has successfully established the effectiveness and versatility of 4-step Quadratic kernel. Some more developments have been done by several authors, [8]-[16].
Cerone, P. (2002). A new Ostrowski type inequality involving integral Means over end intervals. (2), 109-118.
Dragömer, S. S., & Wang, S. (1997). An inequality of Ostr”{u}uski-Gruss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules. (1), 16-20.
Dragömer, S. S., & Wang, S. (1998). Applications of Ostroüski inequality to the estimation of error bounds for some special means and some numerical quadrature rules. , 105-109.
Alomari, M. W. (2012). A companion of Dragomir’s generalization of Ostrowski’s inequality and applications in Numerical Integration. (4), 491-510.
Qayyum, A., Shoaib, M., & Erden, S. (2019). Generalized fractional Ostrowski type inequality for higher order derivatives. (2), 111-124.
Fahad, S., Mustafa, M. A., Ullah, Z., Hussain, T., & Qayyum, A. (2022). Weighted Ostrowski’s Type Integral Inequalities for Mapping Whose First Derivative Is Bounded. , 16-16.
Kashif, A. R., Khan, T. S., Qayyum, A., & Faye, I. (2018). A comparison and error analysis of error bounds. (5), 751-762.
Guessab, A., & Schmeisser, G. (2002). Sharp integral inequalities of the Hermite-Hadamard type. (2), 260-288.
Qayyum, A., Shoaib, M., & Faye, I. (2015). On New Weighted Ostrowski Type Inequalities Involving Integral Means Over End Intervals and Application. (2), 61-67.
Amjad, J., Qayyum, A., Arslan, M., & Fahad, S. (2022). Some new generalized Ostrowski type inequalities with new error bounds. (2), 30-43.
Qayyum, A. (2008). A weighted Ostrowski-Gr”{u}ss type inequality for twice differentiable mappings and applications. , 63-71.
Qayyum, A., Shoaib, M., Matouk, A. E., & Latif, M. A. (2014). On New Generalized Ostrowski Type Integral inequalities. 2014, 1-8.
Qayyum, A., Shoaib, M., & Latif, M. A. (2014). A generalized inequality of ostrowski type for twice differentiable bounded mappings and applications. (38), 1889-1901.
Budak, H., Sarikaya, M. Z. & Qayyum, A. (2021). New refinements and applications of Ostrowski type inequalities for mappings whose nth derivatives are of bounded variation. (2), 424-435.
Nasir, J., Qaisar, S., Butt, S. I., & Qayyum, A. (2022). Some Ostrowski type inequalities for mappings whose second derivatives are preinvex functions via fractional integral operator. (3), 3303-3320.
Iftikhar, M., Qayyum, A., Fahad, S., & Arslan, M. (2021). A new version of Ostrowski type integral inequalities for different differentiable mapping. (1), 353-359.
