Sharp triangle inequality and its reverse in Banach spaces

On Sharp Triangle Inequalities in Banach Spaces II

Sharp constants related to the triangle inequality in Lorentz spaces

The study of the normability of the Lorentz spaces L(R, μ) goes back to the work of G.G. Lorentz [10, 11] (see also [13, 3, 2] for a more recent account of the normability results for the weighted Lorentz spaces). The condition defining these spaces is given in terms of the distribution function and, equivalently, the non-increasing rearrangement of f (see [1] for standard notations and basic d...

Reverses of the Triangle Inequality in Banach Spaces

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.

More on reverse triangle inequality in inner product spaces

Refining some results of Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or complex inner product space (H ;〈·,·〉), r,s > 0, p ∈ (0,s], D = {x ∈ H ,‖rx− sa‖ ≤ p}, x1,x2 ∈D−{0}, and αr,s = min{(r2‖x...

A Sharp Inequality and Its Applications

We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n = 2) leads to a direct proof of Onofri inequality on S.