In the paper, the logarithmically complete monotonicity, logarithmically absolute monotonicity and logarithmically absolute convexity of the function [Γ(1+tx)] [Γ(1+sx)]t for x, s, t ∈ R such that 1 + sx > 0 and 1 + tx > 0 with s 6= t are verified, some known results are generalized.

We consider the problem of Lp-testing of class of bounded derivative properties over hypergrid domain with points distributed according to some product distribution. This class includes monotonicity, the Lipschitz property, (α, β)-generalized Lipschitz and many more properties. Previous results for Lp testing on [n] d for this class were known for monotonicity and c-Lipschitz properties over un...

We show that all negative powers β a,b of the Beta distribution are infinitely divisible. The case b ≤ 1 follows by complete monotonicity, the case b > 1, s ≥ 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a Lévy perpetuity argument involving the hypergeometric series. We also observe that β a,b is self-decomposable if and only if 2a + b + s + bs ≥ 1, and that in this ca...

In this paper, we study the monotonicity properties of the generalized elliptic integrals and establish two Hölder mean inequalities for the generalized Grötzsch ring function μa(r) and the generalized Hersch-Pfluger distortion function φa K(r) . Mathematics subject classification (2010): 30C62, 33E05.