In this paper, some logarithmically completely monotonic, strongly completely monotonic and completely monotonic functions related to the gamma, digamma and polygamma functions are established. Several inequalities, whose bounds are best possible, are obtained.

This paper contains a variety of new integral inequalities for (s,m)-convex functions using Caputo fractional derivatives and Caputo–Fabrizio operators. Various generalizations Hermite–Hadamard-type containing operators are derived those whose (s,m)-convex. Inequalities involving the digamma function special means deduced as applications.

We show that, for both scalar and spinor QED, the two-loop Euler-Heisenberg effective Lagrangian for a constant Euclidean self-dual background has an extremely simple closed-form expression in terms of the digamma function. Moreover, the scalar and spinor QED effective Lagrangians are very similar to one another. These results are dramatic simplifications compared to the results for other backg...

A function f : (0,∞) → R is said to be completely monotonic if (−1)nf (n)(x) ≥ 0 for all x > 0 and n = 0, 1, 2, .... In this paper we present several new classes of completely monotonic functions. Our functions have in common that they are defined in terms of the classical gamma, digamma, and polygamma functions. Moreover, we apply one of our monotonicity theorems to prove a new inequality for ...

In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and luci...

We define a new kind of classical digamma function, and establish some its fundamental identities. Then we apply the formulas obtained, extend tools developed by Flajolet Salvy to study more general Euler-type sums. The main results Salvy's paper (Expo. Math. 7(1) (1998), 15-35) are immediate corollaries in this paper. Furthermore, provide parameterized extensions Ramanujan-type identities that...