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Let H be a complex Hilbert space. For two continuous functions \(f,\) \(g: \left [ a,b\right ] \rightarrow H\) we define the Čebyšev functional \(\displaystyle D\left ( f,g\right ) :=\left ( b-a\right ) \int _{a}^{b}\left \langle f\left ( t\right ) ,g\left ( t\right ) \right \rangle dt-\left \langle \int _{a}^{b}f\left ( t\right ) dt,\int _{a}^{b}g\left ( t\right ) dt\right \rangle . \) In this paper we show among others that if \(f,\) \(g:\left [ a,b\right ] \rightarrow H\) are strongly differentiable functions on the interval \(\left ( a,b\right ) ,\) then \(\displaystyle \begin {aligned}{} \left \vert D\left ( f,g\right ) \right \vert \leq & \frac {1}{2}\left (b-a\right ) \left \Vert f^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }\int _{a}^{b}\left ( b-t\right ) \left ( t-a\right ) \left \Vert g^{\prime }\left ( t\right ) \right \Vert dt \\ \leq & \frac {1}{2}\left ( b-a\right ) ^{3}\left \Vert f^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }\\ &\quad \times \left \{\begin {array}{l} \frac {1}{4}\left \Vert g^{\prime }\right \Vert { }_{\left [ a,b\right ] ,1}, \\ \left ( b-a\right ) ^{1/q}\left [ B\left ( q+1,q+1\right ) \right ]^{1/q}\left \Vert g^{\prime }\right \Vert { }_{\left [ a,b\right ] ,p}, \\ p,q>1,\frac {1}{p}+\frac {1}{q}=1, \\ \frac {1}{6}\left ( b-a\right ) ^{2}\left \Vert g^{\prime }\right \Vert { }_{\left [a,b\right ] ,\infty }, \end {array}\right . \end {aligned} \) where \(\displaystyle \left \Vert h^{\prime }\right \Vert { }_{\left [ a,b\right ] ,p}:=\left (\int _{a}^{b}\left \Vert h^{\prime }\left ( u\right ) \right \Vert ^{p}du\right )^{1/p},\ p\geq 1 \) and \(\left \Vert h^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }:=\sup _{t\in \left ( a,b\right ) }\left \Vert h^{\prime }\left ( u\right ) \right \Vert \) for a strongly differentiable function h on \(\left ( a,b\right ) ,\) while \(B\left ( \cdot ,\cdot \right ) \) is Beta function. Some applications for operator monotone function with examples for power function are also given.

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Several New Inner Product and Norm Inequalities for the Čebyšev Functional in Hilbert Spaces

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  • 发表日期：

    2026年

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  • Silvestru Sever Dragomir

摘要

Let H be a complex Hilbert space. For two continuous functions \(f,\) \(g: \left [ a,b\right ] \rightarrow H\) we define the Čebyšev functional \(\displaystyle D\left ( f,g\right ) :=\left ( b-a\right ) \int _{a}^{b}\left \langle f\left ( t\right ) ,g\left ( t\right ) \right \rangle dt-\left \langle \int _{a}^{b}f\left ( t\right ) dt,\int _{a}^{b}g\left ( t\right ) dt\right \rangle . \) In this paper we show among others that if \(f,\) \(g:\left [ a,b\right ] \rightarrow H\) are strongly differentiable functions on the interval \(\left ( a,b\right ) ,\) then \(\displaystyle \begin {aligned}{} \left \vert D\left ( f,g\right ) \right \vert \leq & \frac {1}{2}\left (b-a\right ) \left \Vert f^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }\int _{a}^{b}\left ( b-t\right ) \left ( t-a\right ) \left \Vert g^{\prime }\left ( t\right ) \right \Vert dt \\ \leq & \frac {1}{2}\left ( b-a\right ) ^{3}\left \Vert f^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }\\ &\quad \times \left \{\begin {array}{l} \frac {1}{4}\left \Vert g^{\prime }\right \Vert { }_{\left [ a,b\right ] ,1}, \\ \left ( b-a\right ) ^{1/q}\left [ B\left ( q+1,q+1\right ) \right ]^{1/q}\left \Vert g^{\prime }\right \Vert { }_{\left [ a,b\right ] ,p}, \\ p,q>1,\frac {1}{p}+\frac {1}{q}=1, \\ \frac {1}{6}\left ( b-a\right ) ^{2}\left \Vert g^{\prime }\right \Vert { }_{\left [a,b\right ] ,\infty }, \end {array}\right . \end {aligned} \) where \(\displaystyle \left \Vert h^{\prime }\right \Vert { }_{\left [ a,b\right ] ,p}:=\left (\int _{a}^{b}\left \Vert h^{\prime }\left ( u\right ) \right \Vert ^{p}du\right )^{1/p},\ p\geq 1 \) and \(\left \Vert h^{\prime }\right \Vert { }_{\left [ a,b\right ] ,\infty }:=\sup _{t\in \left ( a,b\right ) }\left \Vert h^{\prime }\left ( u\right ) \right \Vert \) for a strongly differentiable function h on \(\left ( a,b\right ) ,\) while \(B\left ( \cdot ,\cdot \right ) \) is Beta function. Some applications for operator monotone function with examples for power function are also given.