A Stability Version of Hölder’s Inequality

In the field of geometric inequalities, the expression Bonnesen type is used after Bonnesen clasical refinement of the isoperimetric inequality (cf., for instance, [Os1], [Os2]), where the deviation from the case of equality (the disc) is given in terms of the outer radius and the inradius of a bounded convex body. The term stability type inequality is also used in a related way (cf. [Gr]), meaning that if the deviation from equality is “small”, then the objects under consideration must be “close” to the extremal object. Here we explore the question of how a Bonnesen or stability version of Hölder’s inequality should look like, as we move away from the equality case. Since the functions f and g involved in Hölder’s inequality will usually belong to different spaces, before they can be compared we need to map these functions, with controlled distortion, into a “common measuring ground”. The way we choose to do this is by first normalizing, and then applying the Mazur map from L and L to L. For nonnegative functions in the unit sphere of L the Mazur map into L is defined by f 7→ f . We will be able to utilize its well known properties (cf. for instance, [BeLi]) to obtain useful estimates. As a model for the stability version of Hölder’s inequality, we use the (real) Hilbert space parallelogram identity, suitably rearranged under the assumption that the vectors are nonzero (see (2.0.2) below). With (2.0.2) in mind we obtain a natural, straightforward generalization of the parallelogram identity, valid for 1 < p <∞, though when p 6= 2 equality will of course be lost, cf. (2.2.1). After one has decided which inequality to prove, the argument is standard. In fact, it is the standard argument: From a refined Young’s inequality one obtains a refined Hölder inequality, which in turn entails a refined triangle inequality, which (together with a simple additional observation) yields the uniform convexity of L spaces in the real valued case, with optimal power type estimates for the modulus of convexity. Like the parallelogram identity in the Hilbert space setting, (2.2.1) brings to the fore the geometry of L spaces, and conveys essentially the same information: In order for ‖fg‖1 to be close to ‖f‖p‖g‖q, the angle between the nonnegative L functions |f |p/2 and |g|q/2 must be small, with equality in ‖fg‖1 ≤ ‖f‖p‖g‖q precisely when the angle is zero. Since Hölder’s