The mapping \(\delta_{x,y}\) in normed linear spaces and refinements of the Cauchy-Schwarz inequality
Let \((X,\|\cdot\|)\) be a real normed linear space. The authors introduce the lower and upper semi-inner product by \[ (y,x)_i= \lim_{t\to 0^+}{\|x+ ty\|^2-\|x\|^2\over 2t} \quad\text{and}\quad (y,x)_s= \lim_{t\to 0^+}{\|x+ ty\|^2-\|x\|^2\over 2t}, \] respectively. Studying some geometrical properties of the function \(\delta_{x,y}: \mathbb{R}\to\mathbb{R}\) defined by \(\delta_{x,y}(t)= 2\|x+ ty\|-\|x+2ty\|\) for fixed \(x,y\in X\) they obtain certain refinements of the Cauchy-Schwarz inequality and give some applications to \(L^p(\Omega)\) spaces for \(p> 1\) and spaces \(\ell^1(\mathbb{R})\), \(\ell^2(\mathbb{R})\) and \(c_0(\mathbb{R})\).
scientific article; zbMATH DE number 1472254