The main objective of this paper is an improvement of the original weighted operator arithmetic-geometric mean inequality in Hilbert space. We define the difference operator between the arithmetic and geometric means, and investigate its properties. Due to the derived properties, we obtain a refinement and a converse of the observed operator mean inequality. As an application, we establish one significant operator mean, which interpolates the arithmetic and geometric means, that is, the Heinz operator mean. We also obtain an improvement of this interpolation.
Motivated by the method of interpolating inequalities that makes use of the improved Jensen-type inequalities, in this paper we integrate this approach with the well known Zipf–Mandelbrot law applied to various types of f-divergences and distances, such are Kullback–Leibler divergence, Hellinger distance, Bhattacharyya distance (via coefficient), -divergence, total variation distance and triangular discrimination. Addressing these applications, we firstly deduce general results of the type for the Csiszár divergence functional from which the listed divergences originate. When presenting the analyzed inequalities for the Zipf–Mandelbrot law, we accentuate its special form, the Zipf law with its specific role in linguistics. We introduce this aspect through the Zipfian word distribution associated to the English and Russian languages, using the obtained bounds for the Kullback–Leibler divergence.
In this paper we consider Jessen's functional, defined by means of a positive isotonic linear functional, and investigate its properties. Derived results are then applied to weighted generalized power means, which yields extensions of some recent results, known from the literature. In particular, we obtain the whole series of refinements and converses of numerous classical inequalities such as the arithmetic-geometric mean inequality, Young's inequality and Hölder's inequality
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