Refinements of lower bounds for polygamma functions

Refinements of Lower Bounds for Polygamma Functions

0 te dt (1) for x > 0, the psi function ψ(x) = Γ ′(x) Γ(x) and the polygamma functions ψ (x) for i ∈ N are a series of important special functions and have much extensive applications in many branches such as statistics, probability, number theory, theory of 0-1 matrices, graph theory, combinatorics, physics, engineering, and other mathematical sciences. In [9, Corollary 2], the inequality ψ(x)...

Sharp Inequalities for Polygamma Functions

where μ is a nonnegative measure on [0,∞) such that the integral (2) converges for all x > 0. This means that a function f(x) is completely monotonic on (0,∞) if and only if it is a Laplace transform of the measure μ. The completely monotonic functions have applications in different branches of mathematical sciences. For example, they play some role in combinatorics, numerical and asymptotic an...

Refinements of the Lower Bounds of the Jensen Functional

and Applied Analysis 3 where Pj j ∑ i 1 pi, j 1, . . . , n. 1.7 Lemma 1.6. Let f be a convex function on I, p a positive n-tuple such that Pn ∑n i 1 pi 1 and x1, x2, . . . , xn ∈ I, n ≥ 3 such that x1 ≤ x2 ≤ · · · ≤ xn. For fixed x1, x2, . . . , xk, where k 2, 3, . . . , n− 1, the Jensen functional J x,p, f defined in 1.2 is minimal when xk xk 1 · · · xn−1 xn, that is, J ( x,p, f ) ≥ k−1 ∑ i 1 ...

Some Lower Bounds for Estrada Index

Lower bounds for moments of L-functions.

The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commun. Math. Phys. 214, 57-89 and 91-110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. As an example ...