Abstract
In this paper, the authors establish some inequalities involving the Psi and k-Gamma functions. The procedure utilizes some monotonicity properties of some functions associated with the Psi and k-Gamma functions. Keywords: Gamma function, k-Gamma function, Psi function, Inequality. MSC: 33B15, 26A48.
Highlights
The well-known classical Gamma function, Γ(t) is usually defined for t > 0 by ∞ Γ(t) = e−xxt−1 dx.The p-analogue of the Gamma function is defined for t > 0 and p ∈ N by Γp(t) = t(t +p!pt 1) . . . (t p) t(1 pt t 1
The procedure utilizes some monotonicity properties of some functions associated with the Psi and k-Gamma functions
The psi function, ψ(t) known in literature as the digamma function is defined for t > 0 as the logarithmic derivative of the gamma function
Summary
The q-analogue of the Gamma function is defined (see [4]) for t > 0 and q ∈ (0, 1) by. The k-analogue or the k-Gamma function is defined (see [1]) for t > 0 and k > 0 by. The psi function, ψ(t) known in literature as the digamma function is defined for t > 0 as the logarithmic derivative of the gamma function. The p-analogue, q-analogue and k-analogue of the psi function are equivalently defined for t > 0 as follows. The following series representations for the functions ψ(t) and ψk(t) are valid and are well-known in literature.
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