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

Silvestru Sever Dragomir ()
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Silvestru Sever Dragomir: Victoria University, Applied Mathematics Research Group, ISILC

A chapter in Functional Equations and Ulam’s Problem, 2026, pp 89-111 from Springer

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

Keywords: 46C05; 47A63; 47A99; Hilbert spaces; Integral inequalities; Operator monotone functions (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-08949-6_5

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