Inequalities for Symmetrized or Anti-Symmetrized Inner Products of Complex-Valued Functions Defined on an Interval

For a function f : a , b → ℂ $$f:\left [ a,b\right ] \rightarrow \mathbb {C}$$ we consider the symmetrical transform of f on the interval a , b , $$\left [ a,b\right ],$$ denoted by f̆, and defined by f ̆ t : = 1 2 f t + f a + b − t , t ∈ a , b $$\displaystyle \breve {f}\left ( t\right ) :=\frac {1}{2}\left [ f\left ( t\right ) +f\left ( a+b-t\right ) \right ],t\in \left [a,b\right ] $$ and the anti-symmetrical transform of f on the interval a , b $$\left [ a,b\right ] $$ denoted by f ~ $$\tilde {f}$$ and defined by f ~ : = 1 2 f t − f a + b − t , t ∈ a , b . $$\displaystyle \tilde {f}:=\frac {1}{2}\left [ f\left ( t\right ) -f\left ( a+b-t\right ) \right ] ,t\in \left [ a,b\right ]. $$ We consider in this paper the inner products f , g ⌣ : = ∫ a b f ̆ t ğ t ¯ d t and f , g ∼ : = ∫ a b f ~ t g ~ t ¯ d t , $$\displaystyle \left \langle f,g\right \rangle _{\smile }:=\int _{a}^{b}\breve {f}\left ( t\right ) \overline {\breve {g}\left ( t\right ) }dt\text{ and }\left \langle f,g\right \rangle _{\sim }:=\int _{a}^{b}\tilde {f}\left ( t\right ) \overline { \tilde {g}\left ( t\right ) }dt, $$ the corresponding norms and establish their fundamental properties. Some Schwarz and Grüss’ type inequalities are also provided.