Extended Heinz and Jensen Type Inequalities and Rearrangements

Abstract

In this paper we extend the well known Heinz inequality which says that $2\sqrt {a_{1}a_{2}}\leq H(t) \leq a_{1}+a_{2}$, a ₁, a ₂ > 0, 0 ≤ t ≤ 1, where $H(t)=a_{1}^{t}a_{2}^{1-t}+a_{1}^{1-t}a_{2}^{t}$. We discuss the bounds of H(t) in the intervals t ∈ [1, 2] and t ∈ [2, ∞) using the subquadracity and the superquadracity of φ(x) = x ^t, x ≥ 0 respectively. Further, we extend H(t) to get results related to $\sum _{i=1}^{n}H_{i}(t)=\sum _{i=1}^{n}\left ( a_{i}^{t}a_{i+1}^{1-t}+a_{i}^{1-t}a_{i+1}^{t}\right )$, a _n+1 = a ₁, a _i > 0, i = 1, …, n, where H ₁(t) = H(t). These results, obtained by using rearrangement techniques, show that the minimum and the maximum of the sum $\sum _{i=1}^{n}H_{i}(t)$ for a given t, depend only on the specific arrangements called circular alternating order rearrangement and circular symmetrical order rearrangement of a given set $\left (\mathbf {a}\right ) =\left (a_{1},a_{2},\dots ,a_{n}\right )$, a _i > 0, i = 1, 2, …, n. These lead to extended Heinz type inequalities of $\sum _{i=1}^{n}H_{i}(t)$ for different intervals of t. The results may also be considered as special cases of Jensen type inequalities for concave, convex, subquadratic and superquadratic functions, which are also discussed in this paper.

Dedicated to Lars-Erik Persson on the occasion of his 75th birthday.