The additivity of polygamma functions

Abstract

Abstract. In the note, the functions˛ψ (i) (e x )˛for i ∈ N are proved to be sub-additive on (ln θ i , ∞) and super-additive on (−∞, ln θ i ), where θ i ∈ (0, 1) is the unique root of equation 2˛ψ (i) (θ)˛=˛ψ (i) (θ 2 )˛.

Some properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold

“…For more information on additive and star-shaped functions, please refer to [11,Chapter 16], [13,Sect. 3.4], the papers [2,3,5,7,10,21,22], and closely related references therein. By these relations, we conclude that the reciprocal 1 � (x) is superadditive.…”

Inequalities Involving $q$-Analogue of Multiple Psi Functions

“…The reverse inequality is valid for all positive x and y if and only if α ≤ min(1, c). In [9], B.-N. Guo, F. Qi and Q.-M. Luo discussed the additivity of polygamma functions. In [12] T. Mansour and A. Sh.…”

Generalizations of Several Inequalities Related to Multivariate Geometric Means

“…Inequality (6) reveals that the geometric mean G(x) is sub-additive. For information about the sub-additivity, please refer to [5][6][7][8][9][10][11][12] and the closely related references therein.…”

“…Inequality (6) reveals that the geometric mean G(x) is sub-additive. For information about the sub-additivity, please refer to [5][6][7][8][9][10][11][12] and the closely related references therein. The sub-additive property of the geometric mean G(x) can also be derived from the property that the geometric mean G(x) is a Bernstein function; see [13][14][15][16][17][18][19] and the closely related references therein.…”