Abstract
Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.
Highlights
The Euler gamma function is defined for all positive real numbers x by ∞(x) = tx–1e–t dt.The logarithmic derivative of (x) is called the psi or digamma function
X n(n + x) n=1 where γ = 0.5772 . . . is the Euler–Mascheroni constant, and ψ(m)(x) for m ∈ N are known as the polygamma functions
The gamma, digamma and polygamma functions play an important role in the theory of special functions, and have many applications in other many branches, such as statistics, fractional differential equations, mathematical physics and theory of infinite series
Summary
Some of the work on the complete monotonicity, convexity and concavity, and inequalities of these special functions can be found in [1,2,3,4,5,6, 8, 14,15,16,17, 21, 22, 27,28,29,30, 37,38,39,40,41,42] and the references therein. It is well known that the k-analogues of the digamma and polygamma functions satisfy the following recursive formula and series identities (see [11]): k(x + k) = x k(x), x > 0, (1.1) Some important identities and inequalities involving these functions may be found in [30, 34, 35]. [4], the function φ(x) was proved to be strictly increasing on
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