Scholarlink

Let \(\Phi:I\rightarrow \mathbb {C}\) be an absolutely continuous function on \(\left [m,M\right] \subset \mathring {I}\) , the interior of I and \(h:\left [a,b\right] \rightarrow \left [m,M\right] \) absolutely continuous on \(\left [a,b\right] \) and such that In this chapter, we show among other that, if there exists the constants \(\Gamma >\gamma \) such that condition \(\displaystyle \gamma \leq \Phi ^{\prime}\left (y\right) \leq \Gamma \text{for a.e.}y\in \left [m,M\right] \) holds, then \(\displaystyle \left \vert \frac {1}{b-a}\int _{a}^{b}\left (\Phi \circ h\right) \left (t\right) dt-\Phi \left (\frac {1}{b-a}\int _{a}^{b}h\left (t\right) dt\right) \right \vert \) \(\displaystyle \leq \frac {1}{8}\left (b-a\right) \left (\Gamma -\gamma \right) \left \Vert h^{\prime}\right \Vert {}_{\left [a,b\right],\infty}. \) Moreover, if \(\Phi ^{\prime}\) is absolutely continuous on \(\left [m,M\right] \) with \(\displaystyle \left \Vert \Phi ^{\prime \prime}\right \Vert {}_{\left [m,M\right],\infty}:= \lim func {esssup}_{x\in \left [a,b\right]}\left \vert \Phi ^{\prime \prime}\left (x\right) \right \vert <\infty, \) then also \(\displaystyle \left \vert \frac {1}{b-a}\int _{a}^{b}\left (\Phi \circ h\right) \left (t\right) dt-\Phi \left (\frac {1}{b-a}\int _{a}^{b}h\left (t\right) dt\right) \right \vert \) \(\displaystyle \leq \frac {1}{24}\left (b-a\right) ^{2}\left \Vert h^{\prime}\right \Vert {}_{\left [a,b\right],\infty}^{2}\left \Vert \Phi ^{\prime \prime}\right \Vert {}_{\left [m,M\right],\infty}. \) Applications for the midpoint inequality and for logarithmic inequalities are also given.

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Bounds for the Unweighted Jensen’s Gap of Absolutely Continuous Functions

  • 章节

  • 发表日期：

    2025年

查看全文
  • Silvestru Sever Dragomir

摘要

Let \(\Phi:I\rightarrow \mathbb {C}\) be an absolutely continuous function on \(\left [m,M\right] \subset \mathring {I}\) , the interior of I and \(h:\left [a,b\right] \rightarrow \left [m,M\right] \) absolutely continuous on \(\left [a,b\right] \) and such that In this chapter, we show among other that, if there exists the constants \(\Gamma >\gamma \) such that condition \(\displaystyle \gamma \leq \Phi ^{\prime}\left (y\right) \leq \Gamma \text{for a.e.}y\in \left [m,M\right] \) holds, then \(\displaystyle \left \vert \frac {1}{b-a}\int _{a}^{b}\left (\Phi \circ h\right) \left (t\right) dt-\Phi \left (\frac {1}{b-a}\int _{a}^{b}h\left (t\right) dt\right) \right \vert \) \(\displaystyle \leq \frac {1}{8}\left (b-a\right) \left (\Gamma -\gamma \right) \left \Vert h^{\prime}\right \Vert {}_{\left [a,b\right],\infty}. \) Moreover, if \(\Phi ^{\prime}\) is absolutely continuous on \(\left [m,M\right] \) with \(\displaystyle \left \Vert \Phi ^{\prime \prime}\right \Vert {}_{\left [m,M\right],\infty}:= \lim func {esssup}_{x\in \left [a,b\right]}\left \vert \Phi ^{\prime \prime}\left (x\right) \right \vert <\infty, \) then also \(\displaystyle \left \vert \frac {1}{b-a}\int _{a}^{b}\left (\Phi \circ h\right) \left (t\right) dt-\Phi \left (\frac {1}{b-a}\int _{a}^{b}h\left (t\right) dt\right) \right \vert \) \(\displaystyle \leq \frac {1}{24}\left (b-a\right) ^{2}\left \Vert h^{\prime}\right \Vert {}_{\left [a,b\right],\infty}^{2}\left \Vert \Phi ^{\prime \prime}\right \Vert {}_{\left [m,M\right],\infty}. \) Applications for the midpoint inequality and for logarithmic inequalities are also given.