On Certain Differential Subordination of Harmonic Mean Related to a Linear Function

In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot-Bouquet differential subordination.

Automated summary

This paper studies a certain differential subordination related to the harmonic mean and its symmetry properties, with a focus on the case where the dominant is a linear function. In addition to known results, the paper explores differential subordinations for selected convex functions and the search for the best dominant function or one close to it. The differential subordination of the harmonic mean is a generalization of the Briot-Bouquet differential subordination in this context.