Clarkson inequalities related to convex and concave functions

Abstract

Abstract In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let A 1, …, A n be bounded linear operators on a complex separable Hilbert space H $\mathcal{H}$ and let α 1, …, α n be positive real numbers such that ∑ j = 1 n α j = 1 $\sum\limits^{n}_{j=1}\alpha_{j}=1$ . We show that for every unitarily invariant norm, If f is a non-negative function on [0, ∞) such that f(0) = 0 and g ( t ) = f ( t ) $g(t)=f(\sqrt{t})$ is convex, then | | | ∑ j = 1 n α j f ( | A j | ) | | | ≥ | | | ∑ j , k ∈ S ℓ ( f ( α j α k 4 α ℓ ( 1 − α ℓ ) | A j + A k − 2 ∑ j = 1 n α j A j | ) + f ( α j α k ( 2 α ℓ − 1 ) 4 α ℓ ( 1 − α ℓ ) | A j − A k | ) ) + f ( | ∑ j = 1 n α j A j | ) | | | $$\begin{align*} \bigg|\bigg|\bigg|\sum\limits^{n}_{j=1}\alpha_{j}f(|A_{j}|)\bigg|\bigg|\bigg| &\geq\bigg|\bigg|\bigg|\sum\limits_{j,k\in S_{\ell}}\bigg(f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}}{4\alpha_{\ell}(1-\alpha_{\ell})}}\;\bigg|A_{j}+A_{k}-2\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\\ &\qquad+f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}(2\alpha_{\ell}-1)}{4\alpha_{\ell}(1-\alpha_{\ell})}}|A_{j}-A_{k}|\bigg)\bigg)+f\bigg(\bigg|\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\bigg|\bigg|\bigg| \end{align*}$$ for ℓ = 1, …, n. If f is a non-negative function on [0, ∞) such that g ( t ) = f ( t ) $g(t)=f(\sqrt{t})$ is concave, then the inverse inequality holds. Here, the symbol S ℓ = {1, …, n} ∖ {ℓ} for ℓ ∈ {1, …, n}. In addition, we provide some applications of the above inequalities.