Integral Representations for Bivariate Complex Geometric Mean and Applications

Abstract

Abstract. In the paper, the authors survey integral representations (including the Lévy-Khintchine representations) and applications of some bivariate means (including the logarithmic mean, the identric mean, Stolarsky's mean, the harmonic mean, the (weighted) geometric means and their reciprocals, and the Toader-Qi mean) and the multivariate (weighted) geometric means and their reciprocals, derive integral representations of bivariate complex geometric mean and its reciprocal, and apply these newly-derived integral representations to establish integral representations of Heronian mean of power 2 and its reciprocal.

Monotonicity, convexity and inequalities related to complete $(p,q,r)$-elliptic integrals and generalized trigonometric functions

“…For detailed and more information on means, please refer to [21], [38], [41], [42], [44], and closely related references therein. Let f : I ⊆ (0, ∞) → (0, ∞) be continuous.…”

Double Inequalities for Complete Monotonicity Degrees of Remainders of Asymptotic Expansions of the Gamma and Digamma Functions

“…A notion of completely monotonic degree was invented first in reference [7] and reviewed in the recent paper [20]. It can be used to measure and differentiate complete monotonicity more accurately, and it is also introduced in [7,9,10,11,12,13,16,17,18,19] and closely related references.…”

Determinantal forms and recursive relations of the Delannoy two-functional sequence

“…are known as the rising and falling factorials respectively. The ideas, significance, and reasonability of these generalizations D a,b (k) and D a,b;λ (z) come from the papers [25,34,37,38,39,43,52,53,54,55] and closely related references therein. Inspired by the identity…”